diff --git "a/H9AyT4oBgHgl3EQfffh-/content/tmp_files/load_file.txt" "b/H9AyT4oBgHgl3EQfffh-/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/H9AyT4oBgHgl3EQfffh-/content/tmp_files/load_file.txt" @@ -0,0 +1,545 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf,len=544 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='00341v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='CO] 1 Jan 2023 Symmetric polynomials connecting unsigned and signed relative derangements Ricky Xiao-Feng Chen, Yu-Chen Ruan School of Mathematics, Hefei University of Technology Hefei, Anhui 230601, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' China xiaofengchen@hfut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='cn, 1059568476@qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='com Abstract In this paper, we introduce polynomials (in t) of signed relative derangements that track the number of signed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The polynomials are clearly seen to be in a sence symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Note that relative derangements are those without any signed elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', the evaluations of the polynomials at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Also, the numbers of all signed relative derangements are given by the evaluations at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then the coefficients of the polynomials connect unsigned and signed relative derangements and show how putting elements with signs affects the formation of derangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We first prove a recursion satisfied by these polynomials which results in a recursion satisfied by the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A combinatorial proof of the latter is provided next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We also show that the sequences of the coefficients are unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, other results are obtained, for instance, a kind of dual of a relation between signed derangements and signed relative derangements previously proved by Chen and Zhang is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Keywords: Derangements, Relative derangements, Symmetric polynomials, Uni- modal Mathematics Subject Classifications: 05C05, 05A19, 05A15 1 Introduction A derangement on a set [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', n} is a permutation π = π1π2 · · · πn on [n] such that πi ̸= i for all i ∈ [n], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', a permutation without fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We use Dn to denote the set of derangements on [n] and Dn to denote the number of derangements on [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The study of derangements may date back to Euler who showed that the probability for a random permutation to be a derangement tends to 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is also well known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', Stanley [8, Chapter 2]) that Dn = (n − 1)(Dn−1 + Dn−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (1) 1 A relative derangement π = π1π2 · · ·πn on [n] is a permutation such that πi+1 ̸= πi + 1 for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let Qn denote the set of relative derangements on [n] and Qn = |Qn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' With the aid of the notion of skew derangements, Chen [4] combinatorially showed that Qn = Dn + Dn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (2) A signed permutation π on [n] can be viewed as a bijection on the set [n] �{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' , n} such that π(i) = π(i), where j = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Intuitively, a signed permutation on [n] is just an ordinary permutation π = π1π2 · · · πn with some elements associated with a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, 1342 is a signed permutation on {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' These elements with a bar are called signed elements or bar-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The set of signed permutation on [n] is often denoted by Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A signed derangement [1] on [n] is a signed permutation π = π1π2 · · · πn such that πi ̸= i, for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, 1342 is a signed derangement in B4, whereas 1342 is not since it has a fixed point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A signed relative derangement (or sometimes called relative derangement of type B, see [5]) on [n] is a signed permutation on [n] such that i is not followed by i + 1, and i is not followed by i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, 1324 is a signed relative derangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We denote by DB n and QB n the sets of signed derangements and signed relative derangements on [n], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let DB n = |DB n | and QB n = |QB n |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Based on the notion of signed skew derangements, Chen and Zhang [5] proved that QB n = DB n + DB n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (3) One of our results in this paper is a kind of dual of this relation, that is, we present a relation expressing DB n in terms of fn that counts an essential subset of sequences in QB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Obviously, the subset of sequences with zero signed elements is Qn and hence Qn ⊂ QB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is natural to consider the subset consisting of sequences with m signed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' As such, a polynomial tracking the number of signed elements is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' While many polynomials or q-analogues associated to derangements have been studied, for instance, the q-enumeration of derangements in Bn by flag major index [1], the excedances of derangements [6,10], the q-enumeration of derangements by major index [9], and the cyclic polynomials of derangements [7], our polynomials here seem to have been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In addition, our polynomials have a nice property, namely, they are in a sense symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In Section 2, we introduce the symmetric poly- nomials and prove a recursion satisfied by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Various results are then derived as a consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Section 3 is devoted to presenting a combinatorial proof of the resulting recursion satisfied by the coefficients as well as proving a unimodality property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 2 Symmetric polynomials Let b(π) be the number of signed elements in π ∈ QB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The polynomial of signed relative derangements recording the number of signed elements is then given by QB n (t) = � π∈QB n tb(π) = n � m=0 qn,mtm, 2 where qn,m denotes the number of signed relative derangements with exactly m signed elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is evident that qn,m = qn,n−m as we can obtain a signed relative derangment with n − m bar-elements by turning a signed element into its unsigned counterpart and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, the polynomial QB n (t) is in a sense symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Denote by �QB n the set of signed permutations on the set [n] where in each signed permutation two consecutive entries of the form i(i+ 1) or i(i + 1) for some 1 ≤ i < n−1 appears exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, 4231 ∈ �QB 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For π ∈ QB n , we denote the resulting sequence from removing n or n whichever appears in π by π↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The following lemma should not be hard to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any π ∈ QB n , we have either π↓ ∈ QB n−1 or π↓ ∈ �QB n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Accordingly, we immediately have QB n (t) = � π∈QB n tb(π) = � π∈QB n , π↓∈�QB n−1 tb(π) + � π∈QB n , π↓∈QB n−1 tb(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (4) To obtain a recursion of QB n (t), we next study the two sums on the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (4) in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For π = π1π2 · · · πn−1 ∈ QB n−1 and n ≥ 2, denote by S↑(π) the set of sequences in �QB n that result from π by lifting the elements larger than πi (for some 1 ≤ i ≤ n − 1) by one and replacing πi with a length-two sequence πi(πi + 1), where we define the addition for bar-elements by the rule i + 1 = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, if an element x appears an entry in π, we write x ∈ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3 and any π ∈ QB n , we have � π′∈S↑(π) tb(π′) = b(π)tb(π)+1 + (n − b(π))tb(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any π = π1π2 · · ·πn ∈ QB n , it has b(π) bar-elements and n − b(π) elements without a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any πi ∈ π with a bar, it will generate an additional bar-element after lifting the elements larger than πi (for some 1 ≤ i ≤ n − 1) by one and replacing πi with a length-two sequence πi(πi + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In other words, it will contribute tb(π)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' However, for any πi ∈ π without a bar, the number of bar-elements in the sequence will not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, it contributes tb(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Summarizing the two cases gives the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The lemma right below is not difficult to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If π, π′ ∈ QB n−1 and π ̸= π′, then S↑(π) � S↑(π′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, �QB n = � π∈QB n−1 S↑(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (6) 3 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, we have � π∈QB n , π↓∈�QB n−1 tb(π) = (1 + t) � (t2 − t)QB n−2 ′(t) + (n − 2)QB n−2(t) � , (7) where QB n ′(t) stands for the derivative of QB n (t) with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' First, by construction, there are exactly two signed permutations π, π′ ∈ QB n such that π↓ = π′↓ ∈ �QB n−1, and vice versa, where if n ∈ π then π′ can be obtained by replacing n with n in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thus, tb(π↓) = tb(π′↓) = tb(π) = tb(π′)−1 and � π∈QB n , π↓∈�QB n−1 tb(π) = � π′∈�QB n−1 (1 + t)tb(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Next, we have � π′∈�QB n−1 tb(π′) = � π′′∈QB n−2 � π′∈S↑(π′′) tb(π′) = � π′′∈QB n−2 � b(π′′) · t + � n �� 2 − b(π′′) �� tb(π′′) = � π′′∈QB n−2 � (t − 1)b(π′′)tb(π′′) + (n − 2)tb(π′′)� = (t2 − t)QB n−2 ′(t) + (n − 2)QB n−2(t), where the first two equalities follow from Lemma 3 and Lemma 2, respectively, and then the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, we have � π∈QB n , π↓∈QB n−1 tb(π) = (nt + n − 1)QB n−1(t) + (1 − t) � π′∈QB n−1, n−1∈π′ tb(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A sequence π ∈ QB n where n appears can be clearly obtained by inserting n into a sequence π↓ ∈ QB n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We distinguish two cases: if n − 1 appears in π↓ ∈ QB n−1, there are n − 1 positions where n can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' if n − 1 appears in π↓ ∈ QB n−1, there are n positions where n can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Note that in both cases, we have b(π) = b(π↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thus, � π∈QB n , π↓∈QB n−1, n∈π tb(π) = � π′∈QB n−1, n−1∈π′ (n − 1) · tb(π′) + � π′∈QB n−1, n−1∈π′ n · tb(π′) = (n − 1)QB n−1(t) + � π′∈QB n−1, n−1∈π′ tb(π′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Similarly, the situation of inserting n can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We also distinguish two cases: 4 if n − 1 appears in π↓ ∈ QB n−1, there are n positions where n can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' if n − 1 appears in π↓ ∈ QB n−1, there are n − 1 positions where n can be inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The difference is that in this case, we have b(π) = b(π↓) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thus, � π∈QB n , π↓∈QB n−1, n∈π tb(π) = � π′∈QB n−1,n−1∈π′ nt · tb(π′) + � π′∈QB n−1,n−1∈π′ (n − 1)t · tb(π′) = ntQB n−1(t) − t � π′∈QB n−1,n−1∈π′ tb(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Combining the above two cases, we obtain the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, we have � π′∈QB n−1, n−1∈π′ tb(π′) =(n − 1)tQB n−2(t) + t � (t2 − t)QB n−3 ′(t) + (n − 3)QB n−3(t) � − t � π′′∈QB n−2, n−2∈π′′ tb(π′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Analogously, we first have � π′∈QB n−1, n−1∈π′ tb(π′) = � π′∈QB n−1, n−1∈π′, π′↓∈QB n−2 tb(π′↓) + � π′∈QB n−1, n−1∈π′, π′↓∈�QB n−2 tb(π′↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The first sum of the right-hand side has been obtained in Proposition 5 and equals (n − 1)tQB n−2(t) − t � π′′∈QB n−2, n−2∈π′′ tb(π′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Following the proof of Proposition 4, the second sum of the right-hand side equals � π′∈�QB n−2 t · tb(π′) = t � π′′∈QB n−3 � π′∈S↑(π′′) tb(π′) = t � π′′∈QB n−3 � b(π′′) · t + � n − 3 − b(π′′) �� tb(π′′) = t � π′′∈QB n−3 � (t − 1)b(π′′)tb(π′′) + (n − 3)tb(π′′)� = t � (t2 − t)QB n−3 ′(t) + (n − 3)QB n−3(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The rest is clear and the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Based on Proposition 4–6, we conclude 5 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, the following holds QB n (t) = (n − 1)(t + 1)QB n−1(t) + � (3n − 5)t + (n − 2) � QB n−2(t) + (t3 − t)QB n−2 ′(t) + (2n − 6)tQB n−3(t) + 2t2(t − 1)QB n−3 ′(t), (10) and QB 0 (t) = 0, QB 1 (t) = 1 + t, QB 2 (t) = t2 + 4t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' According to Proposition 4–6, we first obtain QB n (t) = � π∈QB n tb(π) = � π∈QB n , π↓∈�QB n−1 tb(π) + � π∈QB n , π↓∈QB n−1 tb(π) =(1 + t) � (t2 − t)QB n−2 ′(t) + (n − 2)QB n−2(t) � + (nt + n − 1)QB n−1(t) + (1 − t) � π′∈QB n−1 n−1∈π′ tb(π′) =(1 + t) � (t2 − t)QB n−2 ′(t) + (n − 2)QB n−2(t) � + (nt + n − 1)QB n−1(t) + (1 − t) � (n − 1)tQB n−2(t) + t � (t2 − t)QB n−3 ′(t) + (n − 3)QB n−3(t) � − t � π′′∈QB n−2 n−2∈π′↓ tb(π′↓) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Iterating using Proposition 6 and using the fact that � π′∈QB 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 1∈π′ tb(π′) = t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n (t) =(nt + n − 1)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−1(t) + (1 − t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(−1)k+1(n − k)tkQB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−1(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ (−1)n(1 − t)tn−1 + (1 + t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(t2 − t)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='′(t) + (n − 2)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ (1 − t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='�n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(−1)k+1tk� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(t2 − t)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='′(t) + (n − k − 2)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−2(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='=(nt + n − 1)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−1(t) + (2n − 4)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2(t) + (−1)n(1 − t)tn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(−1)ktk−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(n − k − 1) + (2k + 1 − 2n)t + (n − k)t2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−1(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ (t3 − t)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='′(t) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='(−1)k+1tk+1(2t − 1 − t2)QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n−k−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (11) Consequently, we have QB n−1(t) = � (n − 1)t + n − 2 � QB n−2(t) + (2n − 6)QB n−3(t) + (−1)n−1(1 − t)tn−2 + n−3 � k=1 (−1)ktk−1� (n − k − 2) + (2k + 3 − 2n)t + (n − k − 1)t2� QB n−k−2(t) + (t3 − t)QB n−3 ′(t) + n−3 � k=1 (−1)k+1tk+1(2t − 1 − t2)QB n−k−3 ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 6 Then, it is observed that the two sums in the last expression of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (11) equals (−t) � QB n−1(t) − � (n − 1)t + n − 2 � QB n−2(t) − (2n − 6)QB n−3(t) − (−1)n−1(1 − t)tn−2 − (t3 − t)QB n−3 ′(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Plugging it into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (11) and simplifying completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Based on the obtained recursion eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' the first few polynomials of QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='n (t) are com- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='puted and listed below: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='1 (t) =t + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='2 (t) =t2 + 4t + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='3 (t) =3t3 + 14t2 + 14t + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='4 (t) =11t4 + 64t3 + 112t2 + 64t + 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='5 (t) =53t5 + 362t4 + 866t3 + 866t2 + 362t + 53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='6 (t) =309t6 + 2428t5 + 7252t4 + 10300t3 + 7252t2 + 2428t + 309 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='7 (t) =2119t7 + 18806t6 + 66854t5 + 121838t4 + 121838t3 + 66854t2 + 18806t + 2119 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='8 (t) =16687t8 + 165016t7 + 677656t6 + 1497880t5 + 1937368t4 + 1497880t3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ 677656t2 + 165016t + 16687 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='QB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='9 (t) =148329t9 + 1616786t8 + 7513658t7 + 19444106t6 + 30752450t5 + 30752450t4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='+ 19444106t3 + 7513658t2 + 1616786t + 148329 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let F(x, t) = � n≥1 QB n (t)xn be the generating function of QB n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, F(0, t) = 0 and F(x, t) satisfies the following differential equation: ∂F ∂t (x, t) + t + 1 + 3tx + x + 2tx2 t(t2 − 1) + 2t2(t − 1)x ∂F ∂x (x, t) = −1 − t − 2tx t(t2 − 1)x + 2t2(t − 1)x2 − tx2 − 1 t(t2 − 1)x2 + 2t2(t − 1)x3 F(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (12) The proof of Corollary 8 is provided in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Unfortunately, we are unable to solve the differential equation to get explicit formulas for F(x, t) and QB n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let π ∈ QB n be chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, the expectation and variance of the number of signed elements b(π) are respectively E[b(π)] = n 2, Var[b(π)] = Fn + 2n − n2 4 , where Fn satisfies Fn = � (n − 1)2 + (2n − 2)Fn−1 �QB n−1 QB n + � (3n − 2)(n − 2) + (4n − 3)Fn−2 �QB n−2 QB n + � (2n − 2)(n − 3) + (2n − 2)Fn−3 �QB n−3 QB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Recall that qn,m = qn,n−m, and it is easy to see QB n (1) = n � m=0 qn,m, QB n ′(t) = n � m=0 mqn,mtm−1, QB n ′(1) = n � m=0 mqn,m, QB n ′′(t) = n � m=0 m(m − 1)qn,mtm−2, QB n ′′(1) = n � m=0 m(m − 1)qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consequently, we have E[b(π)] = �n m=0 mqn,m �n m=0 qn,m = �n m=0(m + n − m)qn,m/2 �n m=0 qn,m = QB n ′(1) QB n (1) = n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' As for the variance, we compute Var[b(π)] = n� m=0 (m − E[b(π)])2qn,m QB n = n� m=0 m2qn,m + n� m=0 E[b(π)]2qn,m − 2 n� m=0 mE[b(π)]qn,m QB n = n� m=0 � m(m − 1) + m � qn,m + n� m=0 E[b(π)]2qn,m − 2 n� m=0 mE[b(π)]qn,m QB n = QB n ′′(1) + QB n ′(1) + E[b(π)]2QB n (1) − 2E[b(π)]QB n ′(1) QB n (1) = QB n ′′(1) QB n (1) + 2n − n2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' From Theorem 7, we next get QB n ′′(1) = (2n − 2)QB n−1 ′(1) + (2n − 2)QB n−1 ′′(1) + (6n − 4)QB n−2 ′(1) + (4n − 3)QB n−2 ′′(1) + (4n − 4)QB n−3 ′(1) + (2n − 2)QB n−3 ′′(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' By dividing both sides by QB n , the following recurrsion of Fn = QB n ′′(1) QB n (1) can be obtained: Fn = � (n − 1)2 + (2n − 2)Fn−1 �QB n−1(1) QB n (1) + � (3n − 2)(n − 2) + (4n − 3)Fn−2 �QB n−2(1) QB n (1) + � (2n − 2)(n − 3) + (2n − 2)Fn−3 �QB n−3(1) QB n (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 8 The following corollary follows from Theorem 7 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3 and m ≥ 0, we have Qn =(n − 1)Qn−1 + (n − 2)Qn−2, (13) QB n =(2n − 1)QB n−1 + (2n − 4)QB n−2, (14) qn,m =(n − 1)qn−1,m−1 + (n − 1)qn−1,m + (m − 2)qn−2,m−2 + (3n − 5)qn−2,m−1 + (n − m − 2)qn−2,m + (2m − 4)qn−3,m−2 + (2n − 2m − 4)qn−3,m−1, (15) where we make the convention that qn,m = 0 if m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (13) and (14) follow from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (10) by setting t = 0 and t = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (15) is obtained by comparing the coefficients of tm on both sides of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (10) It is easy to see that the case m = 0 of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (15) agrees with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Of course, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (13) and (14) can be also obtained by making use of the recursions satisfied by Dn, DB n , eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (2) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We leave the computation to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the next section, we will present a direct combinatorial proof of the recursion of qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 3 Recursion and unimodality of qn,m The goal of this section is to first prove the recursion of qn,m combinatorially, and then prove the sequence of qn,m is unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Before we proceed, we present a connection to the work of the first author [3] using a slight variation of signed relative derangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Recall the definitions there: Let Γn = {(0, −1), (−1, 0), (1, −2), (−2, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', (n, −n − 1), (−n − 1, n)} be a set of ordered pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For an ordered pair T = (a, b), the element a is called the left entry of T and denoted by T l = a, while b the right entry of T and denoted by T r = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A signed relative derangement (SRD) on Γn is a sequence π = T0T1 · · · Tn such that Ti ∈ Γn, each ordered pair appears at most once in π, (a, b) ∈ Γn and (b, a) ∈ Γn cannot be both contained in π, and for 0 ≤ i ≤ n − 1, T r i ̸= −T l i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This particular form for SRDs was chosen for a reason, as SRDs were also treated as fixed point involutions in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' As such, the first author could provide an upper bound for the number of signed permutations whose reversal distances are maximum possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' An SRD of type 1 on Γn is an SRD π = T0T1T2 · · · Tn such that T0 = (0, −1) and Tn ̸= (n, −n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' An SRD of type 2 on Γn is an SRD π = T0T1T2 · · ·Tn such that T0 = (0, −1) and Tn = (n, −n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let fn and ˆfn denote the number of SRDs of type 1 and type 2 on Γn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Clearly, ˆfn = fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' One of the main results in Chen [3] is the four-term recursion below fn = (2n − 2)fn−1 + (4n − 3)fn−2 + (2n − 2)fn−3, (16) where f1 = 1, f2 = 4, f3 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 9 Following [3], we have known that there is a natural bijection for transfroming SRDs on Γn to the signed relative derangements in the classical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' That is, just view (i, −i−1) as i and (−i−1, i) as i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' But it is worth noting that the condition now becomes that i is not followed by i + 1 and i + 1 is not followed by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Sometimes it is more convenient to use this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For instance, let π[r] denote the sequence obtained from π by reading π reversely (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', right to left) and changing i to i and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, if π is an SRD, then π[r] is also an SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, for an SRD π = 2310, π[r] = 0132 is an SRD too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We refer to π[r] as the conjugate-reverse of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This is not true in the classical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For example, for a signed relative derangement π = 3210, π[r] = 0123 is not a signed relative derangement anymore in the classical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the following, we will use the new version of SRDs if not explicitly stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 3, QB n = (fn + fn−1) + (fn−1 + fn−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The elements π1π2 · · · πn in QB n consist of two classes: π1 = 1 and π1 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The latter is equivalent to SRDs of type 1 and type 2 and counted by fn + ˆfn = fn + fn−1 as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' As for those starting with 1, the subsequence π2 · · ·πn must not start with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is then not hard to see that this class is counted by fn−1 + fn−2, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In view of Lemma 11, the “core” of QB n is really the subset of sequences not starting with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Also, recall that QB n = DB n + DB n−1 obtained by Chen and Zhang [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Accordingly, it suggests the following relation which can be viewed as a dual of this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proposition 12 (Dual of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 2, we have DB n = fn + fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' First, we take the opportunity to present a direct combinatorial proof of a recursion of DB n which is an analogue of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consider signed derangements of length n in DB n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We distinguish the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 1: If 1 appears, it can be placed at any other n − 1 positions except the first position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Suppose 1 is placed at the k-th position for a fixed 1 < k ≤ n, then we consider the elements k and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If k is placed at the first position, the remaining n − 2 entries (other than the first and the k-th entries) could essentially form any signed derangement of length n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, we have DB n−2 signed derangements in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If k is not placed at the first position (note that k could still be placed at the first position), viewing k as 1 (and k as 1), the remaining n − 1 entries other than the k-th entry essentially form a signed derangement of length n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Hence, there are DB n−1 signed derangements in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 10 Since there are n − 1 options for k, we have (n − 1)(DB n−2 + DB n−1) signed derangements where 1 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 2: Consider the case 1 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Clearly, there are DB n−1 signed derangements where 1 is placed at the first position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If 1 is not placed at the first position, in analogy with case 1, we have (n−1)(DB n−2+ DB n−1) such signed derangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Summarizing the above discussion, we have DB n = (2n − 1)DB n−1 + (2n − 2)DB n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (19) Next, let Fn = fn + fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Applying the four-term recurrence eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (16), we have Fn = (2n − 1)fn−1 + (4n − 3)fn−2 + (2n − 2)fn−3 = (2n − 1)Fn−1 + (2n − 2)Fn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' That is, DB n and Fn satisfy the same recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Meanwhile, we have DB 2 = F2 = 5, DB 3 = F3 = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, DB n and fn + fn−1 also have the same initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thus, it is proved that DB n = fn + fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We remark that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (19) can be found in [2], but with a different proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Combining Proposition 12 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (16), we immediately have an alternative proof of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Now we are in a position to prove the recursion eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Let qn,m denote the number of π = π1π2 · · · πn ∈ QB n with m bar-elements and π1 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Equivalently, qn,m counts SRDs of type 1 and 2 on Γn that have m bar-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We first have the following relation which is an analogue of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' qn,m = qn,m + qn−1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (20) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any π ∈ QB n with m bar-elements, π is either in the form π1π2 · · ·πn where π1 ̸= 1 or 1π2 · · ·πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The number of the former is just qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' And the number of the latter is equal to the number of π2 · · · πn where π2 ̸= 2, namely qn−1,m, whence the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the light of Lemma 13, in order for studying qn,m it suffices to study qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' To that end, we generalize the idea for proving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (16) in [3] and obtain Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For n ≥ 4, we have qn,m =(n − 1)qn−1,m + (n − m − 1)qn−2,m + (m − 1)qn−2,m−1 + nqn−1,n−m + (m − 1)qn−2,n−m + (n − m − 1)qn−2,n−m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (21) 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Note that SRDs of type 1 and 2 on Γn with m bar-elements are either in the form 0A11A2 or 0A11A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We will count SRDs in each case separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 1: 0A11A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (i) A2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this case, A1 could essentially be any SRD of length n − 1 with m bar-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' It is easy to see there are qn−1,m + qn−2,m such SRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (ii) A2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consider the induced sequence 1A2A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If there exists no a ∈ [n] such that A2 ends with a while A1 starts with a − 1 or A2 ends with a − 1 while A1 starts with a, then the sequence 1A2A1 could be equivalently any SRD of type 1 or 2 of length n−1 and with m bar-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The latter is counted by qn−1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, there are n − 2 ways to transform each such a sequence into sequences of the form A11A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Hence, there are (n − 2)qn−1,m SRDs lying in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If otherwise, such an a exists, then by construction a ∈ [n] \\ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We claim that for a fixed a ∈ [n] \\ [2], the sequences of the form 1A′ 2aa − 1A′ 1 are in one-to-one correspondence to the SRDs on the set Γn−1 \\ {0, 0} starting with 1 and having m − 1 bar-elements which are counted by qn−2,m−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' the sequences of the form 1A′ 2(a − 1)aA′ 1 are in one-to-one correspondence to the SRDs on the set Γn−1 \\ {0, 0} starting with 1 and having m bar-elements which are counted by qn−2,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The above first case can be seen from replacing aa − 1 with a − 1 and decreasing all other elements greater than a (regardless of if it has a bar) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In particular, this will lose one bar-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The second case can be seen analogously, but without losing a bar-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Conversely, for each of the m − 1 bar-elements in the SRDs on the set Γn−1 \\ {0, 0} starting with 1, say a − 1 (a > 2), we first increase all elements no less than a by one, and then replace a − 1 with aa − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Clearly, the resulting sequence is of the form 1A′ 2aa − 1A′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In addition, there is a unique way to transform such a sequence into an SRD of the form 0A11A2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', 0a − 1A′ 11A′ 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' So, there are (m − 1)qn−2,m−1 SRDs lying in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Analogously, we find there are (n − 2 − m)qn−2,m SRDs of the form 0aA′ 11A′ 2(a − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In summary, for n ≥ 4, the number of SRDs of type 1 and type 2 with m bar-elements on Γn in the form 0A11A2 is given by (n − 1)qn−1,m + (n − m − 1)qn−2,m + (m − 1)qn−2,m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 2: 0A11A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consider the induced sequence 1A[r] 1 A[r] 2 first (Recall A[r] i denotes the conjugate-reverse of Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Apparently, there are n − m bar-elements in A[r] 1 A[r] 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (i) A[r] 1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this scenario, A[r] 2 could essentially be any SRD of length n − 1 with n − m bar- elements the number of which is given by qn−1,n−m + ¯qn−2,n−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (ii) A[r] 1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 12 When A[r] 2 = ∅, 1A[r] 1 is the conjugate-reverse of A11 thus is an SRD of length n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consequently, the number of SRDs in this case is qn−1,n−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Suppose A[r] 2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Similar to case 1 (ii), there are (n − 2)qn−1,n−m SRDs where there is no a ∈ [n] such that A[r] 1 ends with a while A[r] 2 starts with a − 1 or A[r] 1 ends with a − 1 while A[r] 2 starts with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Suppose otherwise such an a exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For a fixed a ∈ [n]/[2], similar to the discussion in case 1 (ii), we claim that the sequences of the form 1A[r] 1 ′aa − 1A[r] 2 ′ are in one-to-one correspondence to the SRDs on the set Γn−1 \\ {0, 0} starting with 1 and having n − m − 1 bar-elements which are counted by (n − m − 1)qn−2,n−m−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' the sequences of the form 1A[r] 1 ′(a−1)aA[r] 2 ′ are in one-to-one correspondence to the SRDs on the set Γn−1 \\ {0, 0} starting with 1 and having n − m bar-elements which are counted by (m − 2)qn−2,n−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In summary, for n ≥ 4, the number of SRDs of type 1 and type 2 with m bar-elements on Γn in the form 0A11A2 is given by nqn−1,n−m + (m − 1)qn−2,n−m + (n − m − 1)qn−2,n−m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Combining the above two cases together, the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Applying Theorem 14, we have qn,m =qn,m + qn−1,m =(n − m − 1)qn−1,m + (n − m − 2)qn−2,m + mqn−1,m + (m − 1)qn−2,m−1 + (m − 1)qn−1,m−1 + (m − 1)qn−2,m−1 + (n − m + 1)qn−1,n−m + (n − m − 1)qn−2,n−m−1 + (n − m)qn−2,n−m−1 + (n − m − 2)qn−3,n−m−2, and qn−1,m−1 =qn−1,m−1 + qn−2,m−1 =(n − m − 1)qn−2,m−1 + (n − m − 2)qn−3,m−1 + (m − 1)qn−2,m−1 + (m−2)qn−3,m−2 + (m−2)qn−2,m−2 + (m−2)qn−3,m−2 + (n−m+1)qn−2,n−m + (n − m − 1)qn−3,n−m−1 + (n − m)qn−3,n−m−1 + (n − m − 2)qn−4,n−m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Summing up the above two equations, we can clear all numbers of the form qx,y and arrive at qn,m + qn−1,m−1 =nqn−1,m−1 + (n − 1)qn−1,m + (m − 2)qn−2,m−2 + (3n − 5)qn−2,m−1 + (n − m − 2)qn−2,m + (2m − 4)qn−3,m−2 + (2n − 2m − 4)qn−3,m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moving qn−1,m−1 to the right-hand side, we obtain eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (15) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 13 Is it true that there will be more signed relative derangements if we turn more unsigned elements into signed elements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Put it differently, is it easier to form a relative derangement if more elements have signs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The answer is apparently negative due to the symmetry of qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' But, how about the cases for m ≤ n/2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This is related to the unimodality of sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' The sequence x0, x1, x2, · · · , xn is said to be unimodal if there exists an index 0 ≤ m ≤ n, called the mode of the sequence, such that x0 ≤ · · · ≤ xm−1 ≤ xm ≥ xm+1 ≥ · · ≥ xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For any fixed n ≥ 1, the sequence qn,0, qn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' , qn,n is unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Thanks to the symmetry of qn,m, it suffices to prove P(n, m) = qn,m − qn,m−1 ≥ 0 for m ≤ n/2, where we still make the convention qn,m = 0 if m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We shall prove this mainly by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' First, from the polynomials of QB n (t) listed in the last section, we observe that for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' , 9 and m ≤ n/2, P(n, m) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Secondly, we claim for any n ≥ 2, P(n, 1) ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' for any n ≥ 4, P(n, 2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In order for proving P(n, 1) ≥ 0 in the case of n ≥ 2, we construct an injection from Qn to QB n,1 (where QB n,i denotes the subset containing signed relative derangements with i bar-elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For each sequence in Qn, replacing n with n, we obtain a unique sequence in QB n,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Obviously, this is an injection and then P(n, 1) ≥ 0 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Analogously, we construct an injection from QB n,1 to QB n,2 for proving P(n, 2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We will classify the sequences in QB n,1 by the largest bar-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 1: If the largest bar-element in QB n,1 is less than n − 1, then we map it to a relative derangement obtained by substituting n for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this case, the obtained relative derangements in QB n,2 have two bar-elements: n and i for some 1 ≤ i < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 2: If the largest bar-element in QB n,1 is exactly n − 1, and n − 1 is not followed by n, then we substitute n for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the case that n − 1 is followed by n, we replace 1 with 1 to obtain a sequence in QB n,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this case, the obtained relative derangements in QB n,2 have two bar-elements: either n − 1 and n, or n − 1 and 1 with an additional feature that n − 1 is followed by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' case 3: Suppose the largest bar-element in QB n,1 is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If n−1 is not followed by n, then we remove the bar of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Meanwhile, we replace n−1 with n − 1 and 1 with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' If n follows n − 1, then we simply replace 1 with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In this case, the obtained relative derangements in QB n,2 have two bar-elements: either n − 1 and 1 with an additional feature that n − 1 is not followed by n, or n and 1 with the feature that n follows n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' In the above mapping procedure, relative derangements in QB n,1 lying in the same case are clearly mapped to distinct relative derangements in QB n,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Moreover, inspecting the patterns of the contained two bar-elements and the additional features, relative de- rangements from different cases are mapped to distinct relative derangements in QB n,2 (for n ≥ 4) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, the above map is indeed an injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Hence, P(n, 2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 14 Now suppose for 1 ≤ n ≤ N and any 0 ≤ m ≤ n/2, P(n, m) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Next, we shall show that P(N + 1, m) ≥ 0 for any 3 ≤ m ≤ (N + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Applying Corollary 10, we first have P(N + 1, m) = qN+1,m − qN+1,m−1 =N(qN,m − qN,m−2) + (N − m − 1)(qN−1,m − qN−1,m−1) + 3(N − 1)(qN−1,m−1 − qN−1,m−2) + (m − 3)(qN−1,m−2 − qN−1,m−3) + 2(N − m − 1)(qN−2,m−1 − qN−2,m−2) + 2(m − 3)(qN−2,m−2 − qN−2,m−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (22) We proceed to distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (i) If 3 ≤ m ≤ (N − 1)/2, we compare the two subscripts of each term qx,y on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (22) and find that y ≤ x/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For instance, since the maximum value of m here is (N − 1)/2, as to qN−1,m−2, we have m − 2 = (N − 5)/2 which satisfies (N − 1)/2 ≥ m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Consequently, qN−1,m−2 −qN−1,m−3 ≥ 0 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Other summands are nonnegative by the same token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, P(N + 1, m) ≥ 0 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (ii) If N/2 ≤ m ≤ (N + 1)/2, m equals either N/2 or (N + 1)/2 since m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' We check the two subscripts of qx,y and find that y > x/2 in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, in the following reasoning, we will make some transformation by the symmetry of qn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' When m = N/2, we replace qN−1,m with qN−1,N−m−1 and regroup the terms on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (22), and obtain P(N + 1, m) =N(qN, N 2 − qN, N−4 2 ) + 3(N − 1)(qN−1, N−2 2 − qN−1, N−4 2 ) + N − 6 2 (qN−1, N−4 2 − qN−1, N−6 2 ) + (N − 2)(qN−2, N−2 2 − qN−2, N−4 2 ) + (N − 6)(qN−2, N−4 2 − qN−2, N−6 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (23) Similarly, when m = (N + 1)/2, we replace qN,m with qN,N−m, qN−1,m with qN−1,N−m−1 and qN−2,m−1 with qN−2,N−m−1 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (22) and regroup the terms to have P(N + 1, m) =N(qN, N−1 2 − qN, N−3 2 ) + 5N − 3 2 (qN−1, N−1 2 − qN−1, N−3 2 ) + N − 5 2 (qN−1, N−3 2 − qN−1, N−5 2 ) + (N − 5)(qN−2, N−3 2 − qN−2, N−5 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (24) Inspecting term by term on the RHS of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (23) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (24), they are all nonnegative by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Therefore, P(N + 1, m) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Acknowledgements The authors would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Yi Wang for pointing out that the roots of QB n (t)’s are not necessarily all real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' A Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='13 In the following, we write ∂F ∂x (x, t) as Fx(x, t) and ∂F ∂t (x, t) as Ft(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then according to 15 the definition of F(x, t), we first have Fx(x, t) = � n≥1 nQB n (t)xn−1, Ft(x, t) = � n≥1 QB n ′(t)xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' For the terms on right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' multiplying by xn and summing over n ≥ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' we respectively obtain � n≥3 (n − 1)tQB n−1(t)xn = tx2 � n≥3 (n − 1)QB n−1(t)xn−2 = tx2(Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − QB 1 (t)) � n≥3 (n − 1)QB n−1(t)xn = x2 � n≥3 (n − 1)QB n−1(t)xn−2 = x2(Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − QB 1 (t)) � n≥3 (t3 − t)QB n−2 ′(t)xn = x2(t3 − t) � n≥3 QB n−2 ′(t)xn−2 = x2(t3 − t)Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) � n≥3 (3n − 5)tQB n−2(t)xn = t � � n≥3 3nQB n−2(t)xn − 5 � n≥3 QB n−2(t)xn� = t � � n≥3 3(n − 2 + 2)QB n−2(t)xn − 5 � n≥3 QB n−2(t)xn� = t � 3 � n≥3 (n − 2)QB n−2(t)xn + 6 � n≥3 QB n−2(t)xn − 5 � n≥3 QB n−2xn� = t � 3x3 � n≥3 (n − 2)QB n−2(t)xn−3 + x2 � n≥3 QB n−2(t)xn−2� = t � 3x3Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + x2F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) � � n≥3 (n − 2)QB n−2(t)xn = x3 � n≥3 (n − 2)QB n−2(t)xn−3 = x3Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) � n≥3 (2t3 − 2t2)QB n−3 ′(t)xn = (2t3 − 2t2)x3 � n≥3 QB n−3 ′(t)xn−3 = (2t3 − 2t2)x3Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) 16 � n≥3 (2n − 6)tQB n−3(t)xn = t � � n≥3 2nQB n−3(t)xn − 6 � n≥3 QB n−3(t)xn� = t � 2 � n≥3 (n − 3 + 3)QB n−3(t)xn − 6 � n≥3 QB n−3(t)xn� = t � 2 � n≥3 (n − 3)QB n−3(t)xn� = t � 2x4 � n≥3 (n − 3)QB n−3(t)xn−4� = 2tx4Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) According to the computation above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' for n ≥ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' we have � n≥3 QB n (t)xn =tx2(Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − QB 1 (t)) + x2(Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − QB 1 (t)) + x2(t3 − t)Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + t � 3x3Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + x2F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) � + x3Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + (2t3 − 2t2)x3Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + 2tx4Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) = � (t + 1)x2 + (3t + 1)x3 + 2tx4� Fx(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + � (t3 − t)x2 + (2t3 − 2t2)x3� Ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) + tx2F(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' t) − (t + 1)2x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Then, F(x, t) is given as follows: F(x, t) =QB 1 (t)x + QB 2 (t)x2 + � n≥3 QB n (t)xn =x + tx + t2x2 + 4tx2 + x2 + � (t + 1)x2 + (3t + 1)x3 + 2tx4� Fx(x, t) + � (t3 − t)x2 + (2t3 − 2t2)x3� Ft(x, t) + tx2F(x, t) − (t + 1)2x2 = � (t + 1)x2 + (3t + 1)x3 + 2tx4� Fx(x, t) + � (t3 − t)x2 + (2t3 − 2t2)x3� Ft(x, t) + tx2F(x, t) + (t + 1)x + 2tx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' After sorting out the above equations, we eventually obtain Ft(x, t) + t + 1 + (3t + 1)x + 2tx2 t(t2 − 1) + 2t2(t − 1)x Fx(x, t) + tx2 − 1 t(t2 − 1)x2 + 2t2(t − 1)x3F(x, t) = −1 − t − 2tx t(t2 − 1)x + 2t2(t − 1)x2, completing the proof of Corollary 8.' 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+page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Liu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Dong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Cyclic derangement polynomials of the wreath product Cr≀Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', 343(12):112109, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Enumerative Combinatorics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Cambridge University Press, Cam- bridge, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Wachs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' On q-derangement numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', 106(1):273–278, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Excedance numbers for the permutations of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' Com- bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=', 20(2):P28, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQfffh-/content/2301.00341v1.pdf'}