M. H. Albert, Micah Coleman, Ryan Flynn, Imre Leader
It is shown that the maximum number of patterns that can occur in a permutation of length $n$ is asymptotically $2^n$. This significantly improves a previous result of Coleman.
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We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern equals (n-2)2^(n-3). We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is (n-3)(n-4)2^(n-5).
We present a method, illustrated by several examples, to find explicit counts of permutations containing a given multiset of three letter patterns. The method is recursive, depending on bijections to reduce to the case of a smaller multiset, and involves a consideration of separate cases according to how the patterns overlap. Specifically, we use the method (i) to provide combinatorial proofs of Bona's formula {2n-3}choose{n-3} for the number of n-permutations containing one 132 pattern and Noonan's formula 3/n {2n}choose{n+3} for one 123 pattern, (ii) to express the number of n-permutations c…
We consider the problem of enumerating the permutations containing exactly $k$ occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an alternative approach to the problem, which yields a proof for a formula which so far only was conjectured (by Noonan and Zeilberger). This approach is based on bijections from permutations to certain lattice paths with ``jumps'', which were first considered by Krattenthaler.
A permutation of n letters is k-prolific if each (n-k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of k-prolific permutations for each k, proving that k-prolific permutations of m letters exist for every m \ge k^2/2+2k+1, and that none exist of smaller size. Key to these results is a natural bijection between k-prolific permutations and certain "permuted" packings of diamonds.
Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the affirmative; in particular, we show that for m > 1, a_(2m)(123) = 10 (2m)!/((m - 2)! (m + 3)!), a_(2m)(321) = 4(m - 2) (2m + 3)!/((m + 1)! (m + 4)!), and a_(2m + 1)(123) = a_(2m + 1)(321) = 3(3m + 4)(m - 1) (2m + 2)!/((m + 1)! (m + 4)!) where a_n(p) is the number of alternating permutations of length n containing the pattern p exactly once.
In this paper, we find an explicit formulas, or recurrences, in terms of generating functions for the cardinalities of the sets $S_n(T;τ)$ of all permutations in $S_n$ that contain $τ\in S_k$ exactly once and avoid a subset $T\subseteq S_3$, $|T|\geq2$. The main body of the paper is divided into three sections corresponding to the cases $|T|=2,3$ and $|T|\geq4$.
Let T_k^m={σ\in S_k | σ_1=m}. We prove that the number of permutations which avoid all patterns in T_k^m equals (k-2)!(k-1)^{n+1-k} for k <= n. We then prove that for any τin T_k^1 (or any τin T_k^k), the number of permutations which avoid all patterns in T_k^1 (or in T_k^k) except for τand contain τexactly once equals (n+1-k)(k-1)^{n-k} for k <= n. Finally, for any τin T_k^m, 2 <= m <= k-1, this number equals (k-1)^{n-k} for k <= n. These results generalize recent results due to Robertson concerning permutations avoiding 123-pattern and containing 132-pattern exactly once.
Numerous versions of the question "what is the shortest object containing all permutations of a given length?" have been asked over the past fifty years: by Karp (via Knuth) in 1972; by Chung, Diaconis, and Graham in 1992; by Ashlock and Tillotson in 1993; and by Arratia in 1999. The large variety of questions of this form, which have previously been considered in isolation, stands in stark contrast to the dearth of answers. We survey and synthesize these questions and their partial answers, introduce infinitely more related questions, and then establish an improved upper bound for one of thes…
We present a characterization of finite permutation groups which contain a transitive dihedral subgroup.
Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= Σ_τλ_f(τ)τ$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(π,R)$ is an occurrence of the permutation pattern $π$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(π,R)$, we have $λ_p(τ) = (-1)^{|τ|-|π|}p^{\star}(τ)$ …
We describe a new method for finding patterns in permutations that produce a given pattern after the permutation has been passed once through a stack. We use this method to describe West-3-stack-sortable permutations, that is, permutations that are sorted by three passes through a stack. We also show how the method can be applied to the bubble-sort operator. The method requires the use of mesh patterns introduced by Brändén and Claesson (2011), as well as a new type of generalized pattern we call a decorated pattern.
Previous work has studied the pattern count on singly restricted permutations. In this work, we focus on patterns of length 3 in multiply restricted permutations, especially for double and triple pattern-avoiding permutations. We derive explicit formulae or generating functions for various occurrences of length 3 patterns on multiply restricted permutations, as well as some combinatorial interpretations for non-trivial pattern relationships.
This paper is continuation of the study of the 1-box pattern in permutations introduced by the authors in \cite{kitrem4}. We derive a two-variable generating function for the distribution of this pattern on 132-avoiding permutations, and then study some of its coefficients providing a link to the Fibonacci numbers. We also find the number of separable permutations with two and three occurrences of the 1-box pattern.
A brief presentation of basic definitions and notation used in permutation patterns research.
We define a variation of Stirling permutations, called quasi-Stirling permutations, to be permutations on the multiset $\{1,1,2,2,\ldots, n,n\}$ that avoid the patterns 1212 and 2121. Their study is motivated by a known relationship between Stirling permutations and increasing ordered rooted labeled trees. We construct a bijection between quasi-Stirling permutations and the set of ordered rooted labeled trees and investigate pattern avoidance for these permutations.
Motivated by study of Mahonian statistics, in 2000, Babson and Steingrimsson introduced the notion of a "generalized permutation pattern" (GP) which generalizes the concept of "classical" permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingrimsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other com…
In this paper we calculate the cardinality of the set S_n(T,tau) of all permutations in S_n that avoid one pattern from S_4 and a nonempty set of patterns from S_3.
We establish a bijection from the set of all permutations (of a given length) that avoid a pattern $q$ and a context-sensitive language.
For an arbitrary finite permutation group $G$, subgroup of the symmetric group $S_\ell$, we determine the permutations involving only members of $G$ as $\ell$-patterns, i.e., avoiding all patterns in the set $S_\ell \setminus G$. The set of all $n$-permutations with this property constitutes again a permutation group. We consequently refine and strengthen the classification of sets of permutations closed under pattern involvement and composition that is due to Atkinson and Beals.
Motivated by the recent proof of the Stanley-Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial resu…
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.
Numerical evidence suggests that certain permutation patterns of length k are easier to avoid than any other patterns of that same length. We prove that these patterns are avoided by no more than (2.25k^2)^n permutations of length n. In light of this, we conjecture that no pattern of length k is avoided by more than that many permutations of length n.
In this paper, we study the occurrence of patterns in the cycle structures of permutations.
We give a solution to a problem posed by Corteel and Nadeau concerning permutation tableaux of length n and the number of occurrences of the dashed pattern 32--1 in permutations on [n]. We introduce the inversion number of a permutation tableau. For a permutation tableau T and the permutation $π$ obtained from T by the bijection of Corteel and Nadeau, we show that the inversion number of T equals the number of occurrences of the dashed pattern 32--1 in the reverse complement of $π$. We also show that permutation tableaux without inversions coincide with L-Bell tableaux introduced by Corteel an…
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…
Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on $n$ letters avoiding $1-3-2$ (or containing $1-3-2$ exactly once) and an arbitrary generalized pattern $τ$ on $k$ letters, or containing $τ$ exactly once. In several cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind, and generating function of Motzkin numbers.
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。
Pattern avoiding machines were recently introduced by Claesson, Ferrari and the current author to gain a better understanding of the classical $2$-stacksort problem. In this paper we generalize these devices by allowing permutations with repeated elements, also known as Cayley permutations. The main result is a description of those patterns such that the corresponding set of sortable permutations is a class. We also show a new involution on the set of Cayley permutations, obtained by regarding a pattern-avoiding stack as an operator. Finally, we analyze two generalizations of pop-stack sorting…
In this paper, we introduce the definitions of simsun succession, simsun cycle succession and simsun pattern. In particular, the ordinary simsun permutations are permutations avoiding simsun pattern 321. We study the descent and peak statistics on permutations avoiding simsun successions. We give a combinatorial interpretation of the q-Eulerian polynomials introduced by Brenti (J. Combin. Theory Ser. A 91 (2000), 137-170). We also present a bijection between permutations avoiding simsun pattern 132 and set partitions.
We investigate permutations and involutions that avoid a pattern of length three and have a {\em unique} longest increasing subsequence.
In 2012, Sagan and Savage introduced the notion of $st$-Wilf equivalence for a statistic $st$ and for sets of permutations that avoid particular permutation patterns which can be extended to generalized permutation patterns. In this paper we consider $inv$-Wilf equivalence on sets of two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns $Π$ and $Π'$ are $inv$-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $Π$ is equal to the generating function for the inversio…
We establish an exact formula for the length of the shortest permutation containing all layered permutations of length $n$, proving a conjecture of Gray.
We consider the problem of packing fixed-length patterns into a permutation, and develop a connection between the number of large patterns and the number of bonds in a permutation. Improving upon a result of Kaplansky and Wolfowitz, we obtain exact values for the expectation and variance for the number of large patterns in a random permutation. Finally, we are able to generalize the idea of bonds to obtain results on fixed-length patterns of any size, and present a construction that maximizes the number of distinct large patterns.
We count the number of occurrences of restricted patterns of length 3 in permutations with respect to length and the number of cycles. The main tool is a bijection between permutations in standard cycle form and weighted Motzkin paths.
We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations avoiding the monotone increasing subsequence of length $k-1$.
We modify the enumeration schemes of Zeilberger and Vatter so that they can efficiently enumerate many new classes of pattern-avoiding permutations including all such classes with a regular insertion encoding.
This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns. We study quadrant marked mesh patterns on up-down and down-up permutations, also known as alternating and reverse alternating permutations, respectively. In particular, we refine classic enumeration results of André on alternating permutations by showing that the distribution of the quadrant marked mesh pattern of interest is given by $(\sec(xt))^{1/x}$ on up-down permutations of even length and by $\int_0^t (\sec(xz))^{1+\frac{1}{x}}dz$ on down-up permutations of odd length.
We present some combinatorial interpretations for coefficients appearing in series partitioning the permutations avoiding 132 along marked mesh patterns. We identify for patterns in which only one parameter is non zero the combinatorial family in bijection with 132-avoiding permutations and also preserving the statistic counted by the marked mesh pattern.
In their study of cyclic pattern containment, Domagalski et al. conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. We show that, for every consecutive pattern $σ$ beginning with 1, the bivariate generating function counting occurrences of $σ$ in circular permutations can be obtained from the generating function counting occurrences of $σ$ in (linear) permutations. This includes all the patterns for which the latter generating function is known.
Call a permutation $k$-inflatable if the sequence of its tensor products with uniform random permutations of increasing lengths has uniform $k$-point pattern densities. Previous work has shown that nontrivial $k$-inflatable permutations do not exist for $k \geq 4$. In this paper, we derive a general formula for the limit densities of patterns in the sequence of tensor products of a fixed permutation with each permutation from a convergent sequence. By applying this result, we completely characterize $3$-inflatable permutations and find explicit examples of $3$-inflatable permutations with vari…
The class of permutations that avoid the bivincular pattern (231, {1},{1}) is known to be enumerated by the Fishburn numbers. In this paper, we call them Fishburn permutations and study their pattern avoidance. For classical patterns of size 3, we give a complete enumerative picture for regular and indecomposable Fishburn permutations. For patterns of size 4, we focus on a Wilf equivalence class of Fishburn permutations that are enumerated by the Catalan numbers. In addition, we also discuss a class enumerated by the binomial transform of the Catalan numbers and give conjectures for other equi…
Pattern expansions of permutation statistics
本文研究了排列统计的模式展开，证明了其有限性并给出了正性的组合准则。
An assortment of problems in permutation patterns: unimodality, equivalence, derangements, and sorting
本文探讨了排列模式中的单峰性、等价性、错排计数及排序问题，提出四个开放研究方向。
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.
A Complete Enumeration of Ballot Permutations Avoiding Sets of Small Patterns
本文完全枚举了避免两个或三个长度为3模式的选票排列，建立了相应的递推关系与公式，并发现了避免132和312的选票排列与Dyck路径左因子间的双射，同时完成了相关的Wilf分类。
We introduce and study a new notion of patterns in Stirling and $k$-Stirling permutations, which we call block patterns. We prove a general result which allows us to compute generating functions for the occurrences of various block patterns in terms of generating functions for the occurrences of patterns in permutations. This result yields a number of applications involving, among other things, Wilf equivalence of block patterns and a new interpretation of Bessel polynomials. We also show how to interpret our results for a certain class of labeled trees, which are in bijection with Stirling pe…
In the last decade a huge amount of articles has been published studying pattern avoidance on permutations. From the point of view of enumeration, typically one tries to count permutations avoiding certain patterns according to their lengths. Here we tackle the problem of refining this enumeration by considering the statistics "first/last entry". We give complete results for every generalized patterns of type $(1,2)$ or $(2,1)$ as well as for some cases of permutations avoiding a pair of generalized patterns of the above types.
We study the problem of counting alternating permutations avoiding collections of permutation patterns including 132. We construct a bijection between the set S_n(132) of 132-avoiding permutations and the set A_{2n + 1}(132) of alternating, 132-avoiding permutations. For every set p_1, ..., p_k of patterns and certain related patterns q_1, ..., q_k, our bijection restricts to a bijection between S_n(132, p_1, ..., p_k), the set of permutations avoiding 132 and the p_i, and A_{2n + 1}(132, q_1, ..., q_k), the set of alternating permutations avoiding 132 and the q_i. This reduces the enumeration…
This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns initiated in "S. Kitaev and J. Remmel, Quadrant marked mesh patterns, J. Integer Sequences 12, Issue 4 (2012), Article 12.4.7.". We study quadrant marked mesh patterns on up-down and down-up permutations.
We show that permutations of size $n$ avoiding both of the dashed patterns 32-41 and 41-32 are equinumerous with indecomposable set partitions of size $n+1$, and deduce a related result.
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…
Enumeration of Dumont permutations avoiding certain four-letter patterns
本文枚举了第四类Dumont排列中避免或包含特定四字母模式的情况，并猜想第一类Dumont排列上两个四字母模式的Wilf等价性。
Pattern-avoiding stabilized-interval-free permutations
本文研究无稳定区间（SIF）排列对长度3经典模式及模式对的避免问题，利用不动点分布、组合论证及与有序森林的双射完成枚举，并提出相关猜想。
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 …
Partially ordered patterns (POPs) generalize the notion of classical patterns studied in the literature in the context of permutations, words, compositions and partitions. In this paper, we give a number of general, and specific enumerative results for POPs in permutations defined by bipartite graphs, substantially extending the list of known results in this direction. In particular, we completely characterize the Wilf-equivalence for patterns defined by the N-shape posets.
A 4AP-free permutation of the positive integers
本文构造了正整数的一个置换，使其不含四项等差数列作为子序列，解决了该组合问题。
We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern counts, both in terms of a permutation and in terms of its image under the fundamental bijection. We use these enumerations to resolve the question of characterizing so-called "shallow" permutations, whose depth (equivalently, disarray/displacement) is minimal with respect to length and reflection length. We present this characterization in several ways, includi…
Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural classes of combinatorial objects which start with either permutations or words and add additional restrictions. These include up-down permutations, generalized Euler permutations, words with no consecutive repeated letters, Young tableaux, and non-backtracking random walks. We develop an extension of the cluster method which we call the {\em generalized clust…
We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method.
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。
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