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.
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 …
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…
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., …
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.
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}.
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…
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.
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…
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…
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…
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…
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.
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.
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…
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 …
We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between the set A_{2n}(1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape (3^n), and between the set A_{2n + 1}(1234) and standard Young tableaux of shape (3^{n - 1}, 2, 1). This represents the first enumeration of alternating permutations avoiding a pattern of length four. We also extend previ…
We study the distribution of the statistics 'number of fixed points' and 'number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions enumerating these two statistics. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3. The main technique is to use bijections between pattern-avoiding permutati…
In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such $n$-permutations are $2^{n-1}$, the number of involutions in $\mathcal{S}_n$, and $2E_n$, where $E_n$ is the $n$-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases. To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form $x-y-z$ (a classical 3-pattern) and beginn…
Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers a structure theorem is given. The structure theorem shows that the class is almost closed under direct sums or has a rational generating function.
We characterize permutations whose Bruhat graphs can be drawn in the plane and those whose Bruhat graphs can be drawn in the torus. In particular, we show these properties are characterized by avoiding finitely many permutations.
Klazar defined and studied a notion of pattern avoidance for set partitions, which is an analogue of pattern avoidance for permutations. Sagan considered partitions which avoid a single partition of three elements. We enumerate partitions which avoid any family of partitions of a 3-element set as was done by Simion and Schmidt for permutations. We also consider even and odd set partitions. We provide enumerative results for set partitions restricted by generalized set partition patterns, which are an analogue of the generalized permutation patterns of Babson and Steingr{\'ı}msson. Finally, in …
We give a permutation pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. In particular, we introduce the notion of a split pattern and show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns 3|12 and 23|1. Continuing, we show that a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties if and only if the corresponding permutation avoids (non-split) patterns 3412, 52341, and 635241.…
Recently, Bóna and Smith defined strong pattern avoidance, saying that a permutation $π$ strongly avoids a pattern $τ$ if $π$ and $π^2$ both avoid $τ$. They conjectured that for every positive integer $k$, there is a permutation in $S_{k^3}$ that strongly avoids $123\cdots (k+1)$. We use the Robinson--Schensted--Knuth correspondence to settle this conjecture, showing that the number of such permutations is at least $k^{k^3/2+O(k^3/\log k)}$ and at most $k^{2k^3+O(k^3/\log k)}$. We enumerate $231$-avoiding permutations of order $3$, and we give two further enumerative results concerning strong …
In this paper we study pattern avoidance in Latin Squares, which gives us a two dimensional analogue of the well studied notion of pattern avoidance in permutations. Our main results include enumerating and characterizing the Latin Squares which avoid patterns of length three and a generalization of the Erdős-Szekeres theorem. We also discuss equivalence classes among longer patterns, and conclude by describing open questions of interest both in light of pattern avoidance and their potential to reveal information about the structure of Latin Squares. Along the way, we show that classical resul…
The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining non-isomorphic principal order ideals is described and applied for small lengths. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, the permutations whose principal order ideals have a form related to boolean posets are also completely described. It is determined when the set of permutations avoiding a particular set of patter…
We demonstrate a natural bijection between a subclass of alternating sign matrices (ASMs) defined by a condition on the corresponding monotone triangle which we call the gapless condition and a subclass of totally symmetric self-complementary plane partitions defined by a similar condition on the corresponding fundamental domains or Magog triangles. We prove that, when restricted to permutations, this class of ASMs reduces to 312-avoiding permutations. This leads us to generalize pattern avoidance on permutations to a family of words associated to ASMs, which we call Gog words. We translate th…
A permutation $π$ contains a pattern $σ$ if and only if there is a subsequence in $π$ with its letters are in the same relative order as those in $σ$. Partially ordered patterns (POPs) provide a convenient way to denote patterns in which the relative order of some of the letters does not matter. This paper elucidates connections between the avoidance sets of a few POPs with other combinatorial objects, directly answering five open questions posed by Gao and Kitaev \cite{gao-kitaev-2019}. This was done by thoroughly analysing the avoidance sets and developing recursive algorithms to derive thes…
In this paper we consider the enumeration of ordered set partitions avoiding a permutation pattern of length 2 or 3. We provide an exact enumeration for avoiding the permutation 12. We also give exact enumeration for ordered partitions with 3 blocks and ordered partitions with n-1 blocks avoiding a permutation of length 3. We use enumeration schemes to recursively enumerate 123-avoiding ordered partitions with any block sizes. Finally, we give some asymptotic results for the growth rates of the number of ordered set partitions avoiding a single pattern; including a Stanley-Wilf type that exhib…
