Andrew Crites
In this paper we study pattern avoidance for affine permutations. In particular, we show that for a given pattern p, there are only finitely many affine permutations in $\widetilde{S}_n$ that avoid p if and only if p avoids the pattern 321. We then count the number of affine permutations that avoid a given pattern p for each p in S_3, as well as give some conjectures for the patterns in S_4.
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In 2017 Davis, Nelson, Petersen, and Tenner pioneered the study of pinnacle sets of permutations and asked whether there exists a class of operations, which applied to a permutation in $\mathfrak{S}_n$, can produce any other permutation with the same pinnacle set and no others. In this paper, we adapt a group action defined by Foata and Strehl to provide a way to generate all permutations with a given pinnacle set. From this we give a closed non-recursive formula enumerating permutations with a given pinnacle set. Thus answering a question posed by Davis, Nelson, Petersen, and Tenner.
The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards from the enumeration of the class, but the simple permutations had not been enumerated explicitly. In particular, the enumeration of the simple skew-merged permutations leads to the first truly structural enumeration of this class as a whole. The extension of this method to a wider collection of classes namely grid classes of infinite paths is discussed.
Enumerating two permutation classes by the number of cycles
本文按循环数对两个置换类 Av_n(312,4321) 和 Av_n(321,4123) 进行枚举，并针对若干统计量细化结果。
Not long ago, Claesson and Mansour proposed some conjectures about the enumeration of the permutations avoiding more than three Babson - Steingr\'ımsson patterns (generalized patterns of type $(1,2)$ or $(2,1)$). The avoidance of one, two or three patterns has already been considered. Here, the cases of four and five forbidden patterns are solved and the exact enumeration of the permutations avoiding them is given, confirming the conjectures of Claesson and Mansour. The approach we use can be easily extended to the cases of more than five forbidden patterns.
Babson and Steingrìmsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1,2) or (2,1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. In the present paper we give a complete solution for the number of permutations avoiding a pair of patterns of type (1,2) or (2,1). We also conjectur…
Motivated by a problem in quantum field theory, we study the up and down structure of circular and linear permutations. In particular, we count the length of the (alternating) runs of permutations by representing them as monomials and find that they can always be decomposed into so-called `atomic' permutations introduced in this work. This decomposition allows us to enumerate the (circular) permutations of a subset of the natural numbers by the length of their runs. Furthermore, we rederive, in an elementary way and using the methods developed here, a result due to Kitaev on the enumeration of…
In this paper, we find an explicit formula for the generating function that counts the circular permutations of length n avoiding the pattern 23 4 1 whose enumeration was raised as an open problem by Rupert Li. This then completes in all cases the enumeration of circular permutations that avoid a single vincular pattern of length four containing one vinculum. To establish our results, we introduce three auxiliary arrays which when taken together refine the cardinality of the class of permutations in question. Rewriting the recurrences of these arrays in terms of generating functions leads to f…
The graph of overlapping permutations is a directed graph that is an analogue to the De Bruijn graph. It consists of vertices that are permutations of length $n$ and edges that are permutations of length $n+1$ in which an edge $a_1\cdots a_{n+1}$ would connect the standardization of $a_1\cdots a_n$ to the standardization of $a_2\cdots a_{n+1}$. We examine properties of this graph to determine where directed cycles can exist, to count the number of directed $2$-cycles within the graph, and to enumerate the vertices that are contained within closed walks and directed cycles of more general lengt…
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of occurrences of arbitrary segmented partially ordered patterns among compositions of (n) with a prescribed number of parts. These patterns generalize the notions of rises, drops, and levels studied in the literature. We also obtain results enumerating parts with given sizes and locations among compositions and palindromic compositions with a given number of part…
In recent work, Zeilberger and the author used a functional equations approach for enumerating permutations with r occurrences of the pattern 12...k. In particular, the approach yielded a polynomial-time enumeration algorithm for any fixed nonnegative r. We extend that approach to patterns of the form 12...(k-2)(k)(k-1) by deriving analogous functional equations and using them to develop similar algorithms that enumerate permutations with r occurrences of the pattern. We also generalize those techniques to handle patterns of the form 23...k1 and derive analogous functional equations and enumer…
Universal cycle for $k$-permutations is a cyclic arrangement in which each $k$-permutation appears exactly once as $k$ consecutive elements. Enumeration problem of universal cycles for $k$-permutations is discussed and one new enumerating method is proposed in this paper. Accurate enumerating formulae are provided when $k=2,3$.
Say that a permutation of $1,2,\ldots,n$ is \textit{$k$-bounded} if every pair of consecutive entries in the permutation differs by no more than $k$. Such a permutation is \textit{anchored} if the first entry is $1$ and the last entry is $n$. We show that the generating function for the enumeration of $k$-bounded anchored permutations is always rational, mirroring the known result on (non-anchored) $k$-bounded permutations due to Avgustinovich and Kitaev. We then explicitly determine the recursive formulas of minimal depth for the number of anchored $k$-bounded permutations of $n$ for $k=2$ an…
In an exercise in the first volume of his famous series of books, Knuth considered sorting permutations by passing them through a stack. Many variations of this exercise have since been considered, including allowing multiple passes through the stack and using different data structures. We are concerned with a variation using pop-stacks that was introduced by Avis and Newborn in 1981. Let $P_k(x)$ be the generating function for the permutations sortable by $k$ passes through a pop-stack. The generating function $P_2(x)$ was recently given by Pudwell and Smith (the case $k=1$ being trivial). We…
Let $dd(I;n)$ denote the number of permutations of $[n]$ with double descent set $I$. For singleton sets $I$, we present a recursive formula for $dd(I;n)$ and a method to estimate $dd(I;n)$. We also discuss the enumeration of certain classes of rim hooks. Let $\mathcal{R}_I(n)$ denote the set of all rim hooks of length $n$ with double descent set $I$, so that any tableau of one of these rim hooks corresponds to a permutation with double descent set $I$. We present a formula for the size of $\mathcal{R}_I(n)$ when $I$ is a singleton set, and we also present a formula for the size of $\mathcal{R…
Enumerating pattern-avoiding permutations by leading terms
本文通过直接计数方法证明了固定领先项的避免模式排列的枚举结果，并研究了具有固定前缀的排列数量，给出了多种模式的精确表达式。
Pattern avoidance for permutations has been extensively studied, and has been generalized to vincular patterns, where certain elements can be required to be adjacent. In addition, cyclic permutations, i.e., permutations written in a circle rather than a line, have been frequently studied, including in the context of pattern avoidance. We investigate vincular pattern avoidance on cyclic permutations. In particular, we enumerate many avoidance classes of sets of vincular patterns of length 3, including a complete enumeration for all single patterns of length 3. Further, several of the avoidance …
Arrow pattern avoidance in permutations: structure and enumeration
本文系统研究了箭头模式避免，证明了结构结果并计数了若干箭头避免类，同时探讨了避免箭头模式对的案例。
We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r=0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n=31. We also extend those functional equations to account for the number of inversions and derive analogous algorithms.
We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $π$ is denoted $Av_P(π)$. We extend a proof of Simion and Schmidt to show that $Av_P(132) \leq Av_P(123)$ for any poset $P$, and we exactly classify the posets for which equality holds.
We consider asymptotics of set partition pattern avoidance in the sense of Klazar. Our main result derives the asymptotics of the number of set partitions avoiding a given set partition within an exponential factor, which leads to a classification of possible growth rates of set partition pattern classes. We further define a notion of permutation-tuple avoidance, which generalizes notions of Aldred et al. and the usual permutation pattern setting, and similarly determine the number of permutation-tuples avoiding a given tuple to within an exponential factor.
We give some results about a bijection associating each permutation with a subexcedant function. This function is related to a particular decomposition of the permutation as a product of transpositions and therefore it has been called transposition array in the literature. In particular, we identify anti-exceedance positions of the permutation through its transposition array and we give an expression of the bijection in terms of the cycle structure of the permutation. We give a characterization of a family of permutations having a non-decreasing transposition array and study length 3 pattern a…
Define a permutation $σ$ to be coprime if $\gcd(m,σ(m)) = 1$ for $m\in[n]$. In this note, proving a recent conjecture of Pomerance, we prove that the number of coprime permutations on $[n]$ is $n!\cdot (c+o(1))^n$ where \[c = \prod_{p\text{ prime }}\frac{(p-1)^{2(1-1/p)}}{p\cdot (p-2)^{(1-2/p)}}.\] The techniques involve entropy maximization for the upper bound, and a mixture of number-theoretic bounds, permanent estimates, and the absorbing method for the lower bound.
Enumerating separable derangements
本研究提出多项式时间算法计算集合[n]的可分错位排列数bₙ，基于生成函数技术，证明bₙ与大Schröder数指数增长常数相同，利用前3000项推测渐近行为，部分回答Vatter的相关问题。
The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of permutation avoidance in cyclic permutations and characterized the avoidance classes for all single permutations of length 4. We continue this work. In particular, we establish a cyclic variant of the Erdos-Szekeres Theorem that any linear permutation of length mn+1 must contain either the increasing pattern of length m+1 or the decreasing pattern of length n+1. We …
We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young t…
The study of pattern avoidance in linear permutations has been an active area of research for almost half a century now, starting with the work of Knuth in 1973. More recently, the question of pattern avoidance in circular permutations has gained significant attention. In 2002-03, Callan and Vella independently characterized circular permutations avoiding a single permutation of size $4$. Building on their results, Domagalski et al. studied circular pattern avoidance for multiple patterns of size $4$. In this article, our main aim is to study circular pattern avoidance of $[4,k]$-pairs, i.e., …
We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern 2143. We use a generating tree approach to construct a recursive bijection between the set A_{2n}(2143) of alternating permutations of length 2n avoiding 2143 and standard Young tableaux of shape (n, n, n) and between the set A_{2n + 1}(2143) of alternating permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux of shape (n + 2, n + 1, n). We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizati…
We give an explicit formula for the number of permutations avoiding cyclically a consecutive pattern in terms of the spectrum of the associated operator of the consecutive pattern. As an example, the number of cyclically consecutive $123$-avoiding permutations in ${\mathfrak S}_{n}$ is given by $n!$ times the convergent series ${\displaystyle \sum_{k=-\infty}^{\infty} \left(\frac{\sqrt{3}}{2π(k+1/3)}\right)^{n}}$ for $n \geq 2$.
We study permutations $p$ such that both $p$ and $p^2$ avoid a given pattern $q$. We obtain a generating function for the case of $q=312$ (equivalently, $q=231$), we prove that if $q$ is monotone increasing, then above a certain length, there are no such permutations, and we prove an upper bound for $q=321$. We also present some intriguing questions in the case of $q=132$.
In this paper, we study pattern avoidances of generalized permutations and show that the number of all generalized permutations avoiding $π$ is independent of the choice of $π\in S_3$, which extends the classic results on permutations avoiding $π\in S_3$. Extending both Dyck path and Riordan path, we introduce the Catalan-Riordan path which turns out to be a combinatorial interpretation of the difference array of Catalan numbers. As applications, we interpret both Motzkin and Riordan numbers in two ways, via semistandard Young tableaux of two rows and generalized permutations avoiding $π\in S_…
We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $k\in\mathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Guðmundsson. Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called \emph{$2$-avoidance}.
Circular permutations on {1,2,...,n} that avoid a given pattern correspond to ordinary (linear) permutations that end with n and avoid all cyclic rotations of the pattern. Three letter patterns are all but unavoidable in circular permutations and here we give explicit formulas for the number of circular permutations that avoid one four letter pattern. In the three essentially distinct cases, the counts are as follows: the Fibonacci number F_{2n-3} for the pattern 1324, 2^{n-1}-(n-1) for 1342, and 2^{n}+1-2n-{n}choose{3} for 1234.
In recent years, there has been increasing interest in consecutive pattern avoidance in permutations. In this paper, we introduce two approaches to counting permutations that avoid a set of prescribed patterns consecutively. These algoritms have been implemented in the accompanying Maple package CAV, which can be downloaded from the author's website. As a byproduct of the first algorithm, we have a theorem giving a sufficient condition for when two pattern sets are strongly (consecutively) Wilf-Equivalent. For the implementation of the second algorithm, we define the cluster tail generating fu…
Enumerating Permutations Avoiding Split Patterns 3|12 and 23|1
本文针对避开分裂模式 3|12 和 23|1 的排列给出计数公式，并将其与 Schubert 簇上的纤维丛结构及修正 Bessel 函数联系起来。
We consider the set of affine permutations that avoid a fixed permutation pattern. Crites has given a simple characterization for when this set is infinite. We find the generating series for this set using the Coxeter length statistic and prove that it can always be represented as a rational function. We also give a characterization of the patterns for which the coefficients of the generating series are periodic. The proofs exploit a new polyhedral encoding for the affine symmetric group.
Staircase diagrams, pattern avoidance, and smooth permutations
本文研究阶梯图偏序结构与光滑置换模式避免的联系，并枚举了包括抛光置换在内的若干光滑置换子类。
For a set of permutation patterns $Π$, let $F^\text{st}_n(Π,q)$ be the st-polynomial of permutations avoiding all patterns in $Π$. Suppose $312\inΠ$. For a class of permutation statistics which includes inversion and descent statistics, we give a formula that expresses $F^\text{st}_n(Π;q)$ in terms of these st-polynomials where we take some subblocks of the patterns in $Π$. Using this formula, we can construct many examples of nontrivial st-Wilf equivalences. In particular, this disproves a conjecture by Dokos, Dwyer, Johnson, Sagan, and Selsor that all $\text{inv}$-Wilf equivalences are trivi…
We consider asymptotics of set partition pattern avoidance in the sense of Klazar. One of the results of this paper extends work of Alweiss, and finds a classification for set partitions $π$ such that the number of set partitions of $[n]$ avoiding $π$ grows more slowly than $n^{cn}$ for all $c>0$. Several conjectures are proposed, and the related question of asymptotics of parallel ($k$-tuple) permutation pattern avoidance is considered and solved completely to within an exponential factor, generalizing Marcus and Tardos's 2004 proof of the Stanley-Wilf Conjecture.
Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A partial permutation of length n with k holes is a sequence of symbols $π= π_1π_2 ... π_n$ in which each of the symbols from the set {1,2,...,n-k} appears exactly once, while the remaining k symbols of $π$ are "holes". We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length k co…
Pattern avoidance in non-crossing and non-nesting permutations
本文针对非交叉与非嵌套排列中长度为3的单模式避免问题，给出了避免模式231（及其对称模式132、213、312）的排列的生成函数，解决了此前未决的计数问题。
A permutation whose any prefix has no more descents than ascents is called a ballot permutation. In this paper, we present a decomposition of ballot permutations that enables us to construct a bijection between ballot permutations and odd order permutations, which proves a set-valued extension of a conjecture due to Spiro using the statistic of peak values. This bijection also preserves the neighbors of the largest letter in permutations and thus resolves a refinement of Spiro' s conjecture proposed by Wang and Zhang. Our decomposition can be extended to well-labelled positive paths, a class o…
Max-tree for d-permutations and pattern avoidance
本文研究了d-排列与模式避免之间的关系，提出了一种将d-排列映射到d-ary树的算法，解决了Bonichon和Morel提出的猜想。
Pattern avoidance in nonnesting permutations
研究非嵌套排列中避免不同长度模式的计数问题，推导闭合公式和生成函数，涉及卡特兰和斐波那契数的出现。
We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the generating function explicitly, for integer compositions of n that avoid a given pattern of length 3 and we show that the answer is the same for all such patterns. We also show that the number of multiset permutations that avoid a given three-letter pattern is the same for all such patterns, thereby extending and refining earlier results of Albert, Aldred et al., and by Atkinson, Walker and Linton. Further…
Pattern avoidance in permutations and their rotations
研究排列及其旋转中的模式避免问题，给出前\(k\geq2\)次旋转都避免给定长度为三的模式的排列数等精确公式，所研究的威尔夫等价类由补和反转决定，还对前两次旋转避免不同模式的排列进行分类枚举。
Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors have shown that Patience Sorting provides an algorithmic description for permutations avoiding the barred (generalized) permutation pattern $3-\bar{1}-42$. Motivated by this and a recently formulated geometric form for Patience Sorting in terms of certain intersecting lattice paths, we study the related themes of restricted input and avoidance of similar barred permutation patterns. One such result is to characterize those permutat…
We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size $N$ that avoid the monotone decreasing pattern of fixed size $m$. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $N\to\infty$. For instance, the number of bounded affine permutations of size $N$ that avoid $321$ is asymptotically equal to $4^N (N/4π)^{1/2}$. We also prove a permuton-like result for the…
On $d$-permutations and Pattern Avoidance Classes
本文研究了多维排列（$d$-排列）的模式避免类，通过证明四个猜想扩展了前人工作，穷举枚举了避免任意两个或三个大小为 $3$ 的固定模式的 $3$-排列，建立了相应的递推关系，并为 OEIS 数据库贡献了新序列。
In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.
In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the $n$-th large Schröde…
We construct an intriguing bijection between $021$-avoiding inversion sequences and $(2413,4213)$-avoiding permutations, which proves a sextuple equidistribution involving double Eulerian statistics. Two interesting applications of this result are also presented. Moreover, this result inspires us to characterize all permutation classes that avoid two patterns of length $4$ whose descent polynomial equals that of separable permutations.
Recently, Babson and Steingrimsson have introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider pattern avoidance for such patterns, and give a complete solution for the number of permutations avoiding any single pattern of length three with exactly one adjacent pair of letters. We also give some results for the number of permutations avoiding two different patterns. Relations are exhibited to several well studied combinatorial structures, such as set partitions, Dyck paths, Motzkin paths, an…
The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213- and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.
Pattern avoidance in compositions and powers of permutations
本文定义正整数合成的模式避免概念，并利用它枚举避免链 (312,321:σ) 的长度为 n 的排列，其中 σ 为任意模式，同时枚举避免链 (312,4321:σ) 的排列，其中 σ 为 S_3 中的模式。
This article introduces an analogue of permutation classes in the context of polyominoes. For both permutation classes and polyomino classes, we present an original way of characterizing them by avoidance constraints (namely, with excluded submatrices) and we discuss how canonical such a description by submatrix-avoidance can be. We provide numerous examples of permutation and polyomino classes which may be defined and studied from the submatrix-avoidance point of view, and conclude with various directions for future research on this topic.
On a question about pattern avoidance of cyclic permutations
本文通过分析循环形式结构和应用Dilworth定理，解决了循环排列中避免特定模式τ=1432的问题。
Babson and Steingr\'ımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider n-permutations that avoid the generalized pattern 1-32 and whose k rightmost letters form an increasing subword. The number of such permutations is a linear combination of Bell numbers. We find a bijection between these permutations and all partitions of an $(n-1)$-element set with one subset marked that satisfy certain additional conditions. Also we find the e.g.f. for the number of permutations that avoid a g…
A factorization of a permutation into transpositions is called "primitive" if its factors are weakly ordered. We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described.
In [BabStein] Babson and Steingr\'ımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper we find the exponential generating function for the remaining case…
We study questions of even-Wilf-equivalence, the analogue of Wilf-equivalence when attention is restricted to pattern avoidance by permutations in the alternating group. Although some Wilf-equivalence results break when considering even-Wilf-equivalence analogues, we prove that other Wilf-equivalence results continue to hold in the even-Wilf-equivalence setting. In particular, we prove that t(t-1)...321 and (t-1)(t-2)...21t are even-shape-Wilf-equivalent for odd t, paralleling a result (which held for all t) of Backelin, West, and Xin for shape-Wilf-equivalence. This allows us to classify the …
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