We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The counting sequences for the two representative three-letter patterns, 321 and 132, have respective generating functions (1+x^2)(C(x)-1)/(1+x+x^2-x C(x)) and C(x+x^3) where C(x) is the generating function for the Catalan numbers.
This paper completes a project to enumerate permutations avoiding a triple T of 4-letter patterns, in the sense of classical pattern avoidance, for every T. There are 317 symmetry classes of such triples T and previous papers have enumerated avoiders for all but 14 of them. One of these 14 is conjectured not to have an algebraic generating function. Here, we find the generating function for each of the remaining 13, and it is algebraic in each case.
Let B_n be the hyperoctahedral group; that is, the set of all signed permutations on n letters, and let B_n(T) be the set of all signed permutations in B_n which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets B_n(T) where $T \subseteq B_2$. This allow us to express these cardinalities via inverse of binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.
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 $|T|\geq4$.
We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation $τ$ on k letters, or containing $τ$ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.
Recently, Grytczuk, Kordulewski, and Niewiadomski defined an extremal word over an alphabet $\mathbb{A}$ to be a word with the property that inserting any letter from $\mathbb{A}$ at any position in the word yields a given pattern. In this paper, we determine the number of extremal $XY_1XY_2X\dots XY_tX$-avoiding words on a $k$-letter alphabet. We also derive a lower bound on the shortest possible length of an extremal square-free word on a $k$-letter alphabet that grows exponentially in $k$.
This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is show…
We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain patterns and provide bijective proofs for the resulting formulas.
In 1990 West conjectured that there are $2(3n)!/((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on $n$ letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation $τ$ on $k$ letters. In several interesting cases this generating function can be expressed in terms of the generating f…
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence …
We construct an infinite word $w$ over the $5$-letter alphabet such that for every factor $f$ of $w$ of length at least two, there exists a cyclic permutation of $f$ that is not a factor of $w$. In other words, $w$ does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [TCS 2018]
In [GM] Guibert and Mansour studied involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern on k letters. They also established a bijection between 132-avoiding involutions and Dyck word prefixes of same length. Extending this bijection to bilateral words allows to determine more parameters; in particular, we consider the number of inversions and rises of the involutions onto the words. This is the starting point for considering two different directions: even/odd involutions and statistics of some generalized patterns. T…
We find exact formulas and/or generating functions for the number of words avoiding 3-letter generalized multipermutation patterns and find which of them are equally avoided.
Building an infinite square-free word by appending one letter at a time while simultaneously avoiding the creation of squares is most likely to fail. When the alphabet has two letters this approach is impossible. When the alphabet has three or more letters, one will most probably create a word in which the addition of any letter invariably creates a square. When one restricts the set of undesired squares to a finite one, this can be possible. We study the constraints on the alphabet and the set of squares which permit this approach to work.
We say that a word $w$ on a totally ordered alphabet avoids the word $v$ if there are no subsequences in $w$ order-equivalent to $v$. In this paper we suggest a new approach to the enumeration of words on at most $k$ letters avoiding a given pattern. By studying an automaton which for fixed $k$ generates the words avoiding a given pattern we derive several previously known results for these kind of problems, as well as many new. In particular, we give a simple proof of the formula \cite{Reg1998} for exact asymptotics for the number of words on $k$ letters of length $n$ that avoids the pattern …
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…
A permutation $π$ is said to be {\em Dumont permutations of the first kind} if each even integer in $π$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $π$ (see, for example, \cite{Z}). In \cite{D} Dumont showed that certain classes of permutations on $n$ letters are counted by the Genocchi numbers. In particular, Dumont showed that the $(n+1)$st Genocchi number is the number of Dummont permutations of the first kind on $2n$ letters. In this paper we study the number of Dumont permutations of the first kind on $n$ let…
The large Schroder numbers are known to count several classes of permutations avoiding two 4-letter patterns. Here we show they count another family of permutations, those whose left to right minima decomposition, when reversed, is 321-avoiding. The main tool is the Mansour-Deng-Du bijection from 321-avoiding permutations to Dyck paths.
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…
Two finite words $u,v$ are 2-binomially equivalent if, for all words $x$ of length at most 2, the number of occurrences of $x$ as a (scattered) subword of $u$ is equal to the number of occurrences of $x$ in $v$. This notion is a refinement of the usual abelian equivalence. A 2-binomial square is a word $uv$ where $u$ and $v$ are 2-binomially equivalent. In this paper, considering pure morphic words, we prove that 2-binomial squares (resp. cubes) are avoidable over a 3-letter (resp. 2-letter) alphabet. The sizes of the alphabets are optimal.
We consider avoiding squares and overlaps over the natural numbers, using a greedy algorithm that chooses the least possible integer at each step; the word generated is lexicographically least among all such infinite words. In the case of avoiding squares, the word is 01020103..., the familiar ruler function, and is generated by iterating a uniform morphism. The case of overlaps is more challenging. We give an explicitly-defined morphism phi : N* -> N* that generates the lexicographically least infinite overlap-free word by iteration. Furthermore, we show that for all h,k in N with h <= k, the…
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…
There are several versions of permutation pattern avoidance that have arisen in the literature, and some known examples of two different types of pattern avoidance coinciding. In this paper, we examine barred patterns and vincular patterns. Answering a question of Steingrimsson, we determine when barred pattern avoidance coincides with avoiding a finite set of vincular patterns, and when vincular pattern avoidance coincides with avoiding a finite set of barred patterns. There are 720 barred patterns with this property, each having between 3 and 7 letters, of which at most 2 are barred, and the…
We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation $τ$ on k letters, or containing $τ$ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.
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…
We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding ce…
We study generating functions for the number of involutions in $S_n$ avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation $τ$ on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and $12... k$ have the same enumerative formula according to the length than involutions avoiding both 132 and any {\em double-wedge pattern} possibly followed by fixed points of total length $k$. Many results are also…
Following Ekhad and Zeilberger (The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, Dec 5 2014; see also arXiv:1412.2035), we study the asymptotics for large $n$ of the number $A_{d,r}(n)$ of words of length $rn$ having $r$ letters $i$ for $i=1..n$, and having no increasing subsequence of length $d$. We prove an asymptotic formula conjectured by these authors, and we give explicitly the multiplicative constant appearing in the result, answering a question they asked. These two results should make the OEIS richer by 100+25=125 dollars. In the case $r=1$ we recover Regev's result for …
We give several modifications of the Goulden-Jackson Cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various 'weights' on words, including single letter probability distributions, double letter (i.e. pairwise) probability distributions, and triple letter probability distributions. We also describe an alternative, recursive approach to finding such generating functions. We describe Maple implementations of the various modifications. The accompanying Maple package is available at th…
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.
Recently, Yan and the first named author investigated systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length $3$, where two of the three letters are required to be adjacent. They established many connections with familiar combinatorial families and proposed several interesting conjectures. The objective of this paper is to address two of their conjectures concerning the enumeration of $\underline{12}0$-avoiding inversion or ascent sequences.
A 321-k-gon-avoiding permutation pi avoids 321 and the following four patterns: k(k+2)(k+3)...(2k-1)1(2k)23...(k+1), k(k+2)(k+3)...(2k-1)(2k)123...(k+1), (k+1)(k+2)(k+3)...(2k-1)1(2k)23...k, (k+1)(k+2)(k+3)...(2k-1)(2k)123...k. The 321-4-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincare polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with consta…
A permutation is called layered if it consists of the disjoint union of substrings (layers) so that the entries decrease within each layer, and increase between the layers. We find the generating function for the number of permutations on $n$ letters avoiding $(1,2,3)$ and a layered permutation on $k$ letters. In the most interesting case of two layers, the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind.
We show that the number of members of S_n avoiding any one of five specific triples of 4-letter patterns is given by sequence A111279 in OEIS, which is known to count weak sorting permutations. By numerical evidence, there are no other (non-trivial) triples of 4-letter patterns giving rise to this sequence. We make use of a variety of methods in proving our result, including recurrences, the kernel method, direct counting, and bijections.
In this paper we study graphs defined by pattern-avoiding words. Word-representable graphs have been studied extensively following their introduction in 2000 and are the subject of a book published by Kitaev in 2015. Recently there has been interest in studying graphs represented by pattern-avoiding words. In particular, in 2016, Gao, Kitaev, and Zhang investigated 132-representable graphs, that is, word-representable graphs that can be represented by a word which avoids the pattern 132. They proved that all 132- representable graphs are circle graphs and provided examples and properties of 13…
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…
Enumeration problems related to words avoiding patterns as well as permutations that contain the pattern $123$ exactly once have been studied in great detail. However, the problem of enumerating words that contain the pattern $123$ exactly once is new and will be the focus of this paper. Previously, Doron Zeilberger provided a shortened version of Alexander Burstein's combinatorial proof of John Noonan's theorem that the number of permutations with exactly one $321$ pattern is equal to $\frac{3}{n} \binom{2n}{n+3}$. Surprisingly, a similar method can be directly adapted to words. We are able t…
Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In this paper we study the generating functions for the number of permutations on $n$ letters avoiding a generalized pattern $ab\mn c$ where $(a,b,c)\in S_3$, and containing a prescribed number of occurrences of generalized pattern $cd\mn e$ where $(c,d,e)\in S_3$. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.
For a permutation $π$, let $S_{n}(π)$ be the number of permutations on $n$ letters avoiding $π$. Marcus and Tardos proved the celebrated Stanley-Wilf conjecture that $L(π)= \lim_{n \to \infty} S_n(π)^{1/n}$ exists and is finite. Backed by numerical evidence, it has been conjectured by many researchers over the years that $L(π)=Θ(k^2)$ for every permutation $π$ on $k$ letters. We disprove this conjecture, showing that $L(π)=2^{k^{Θ(1)}}$ for almost all permutations $π$ on $k$ letters.
Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimss…
We consider avoidance of permutation patterns with designated gap sizes between pairs of consecutive letters. We call the patterns having such constraints distant patterns (DPs) and we show their relation to other pattern notions investigated in the past. New results on DPs with 2 and 3 letters are obtained. Furthermore, we show how one can use DPs to prove two former conjectures of Kuszmaul without a computer. In addition, we deduce a surprising relation between the sets of permutations avoiding the classical patterns $123$ and $132$ by looking at a class of DPs with tight gap constraints. So…
In this article, dedicated with admiration and gratitude to guru Neil Sloane on his 75-th birthday, we observe that the generating functions for multi-set permutations that do not contain an increasing subsequence of length d, and where every letter appears the same number of times, say r, are always D-finite, (for every d and every r), and we actually crank out the first few terms of quite a few of them, many of whom are not yet in the OEIS. We also state a conjectured asymptotic formula for these sequences, that reduces to Amitai Regev's famous formula when r=1, and pledge a 100 dollar donat…
Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following \cite{BCS}, let $e_kπ$ (respectively; $f_kπ$) be the number of the occurrences of the generalized pattern $12\mn3\mn...\mn k$ (respectively; $21\mn3\mn...\mn k$) in $π$. In the present note, we study the distribution of the statistics $e_kπ$ and $f_kπ$ in a permutation avoiding the classical pattern $1\mn3\mn2$. Also we present an applications, which relates the Narayana numbers, Catalan …
Recently, Kitaev [Ki2] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingrímsson [BS]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, extend and generalize several known results, as well as get some new results. In particular, we give the g.f. for the entire distribution of the maximum number of non-overlapping occurrences of a pattern p…
In [BS] Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let $f_{τ;r}(n)$ be the number of $1\mn3\mn2$-avoiding permutations on $n$ letters that contain exactly $r$ occurrences of $τ$, where $τ$ a generalized pattern on $k$ letters. Let $F_{τ;r}(x)$ and $F_τ(x,y)$ be the generating functions defined by $F_{τ;r}(x)=\sum_{n\geq0} f_{τ;r}(n)x^n$ and $F_τ(x,y)=\sum_{r\geq0}F_{τ;r}(x)y^r$. We find an explicit expression for $F_τ(x,y)$ in the form of a continued fraction for whe…
In this paper, we study the Wilf-type equivalence relations among multiset permutations. We identify all multiset equivalences among pairs of patterns consisting of a pattern of length three and another pattern of length at most four. To establish our results, we make use of a variety of techniques, including Ferrers-equivalence arguments, sorting by minimal/maximal letters, analysis of active sites and direct bijections. In several cases, our arguments may be extended to prove multiset equivalences for infinite families of pattern pairs. Our results apply equally well to the Wilf-type classif…
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…
Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for k-ary words involving vincular patterns containing a single dash, which explain the majority of the equivalences witnessed for such patterns of length four. When combined with previous results, numerical evidence, and some arguments in specific cases, we obtain the complete Wilf-classification for all vincular patterns of length four containing a single dash. In some cases, our proof shows further t…
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…
Let f_n^r(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12... k, and let F_r(x;k) and F(x,y;k) be the generating functions defined by $F_r(x;k)=\sum_{n\gs0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\gs0}F_r(x;k)y^r$. We find an explcit expression for F(x,y;k) in the form of a continued fraction. This allows us to express F_r(x;k) for $1\ls r\ls k$ via Chebyshev polynomials of the second kind.
The notion of a word-representable graph has been studied in a series of papers in the literature. A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. If $V =\{1, \ldots, n\}$, this is equivalent to saying that $G$ is word-representable if for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if the subword $w_{\{x,y\}}$ of $w$ consisting of all occurrences of $x$ or $y$ in $w$ has no consecutive occurrence of the pattern 11. In this paper, we introduce the study of $…
Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to (2+2)-free posets and various other combinatorial structures. We study pattern avoidance in ascent sequences, giving several results for patterns of lengths up to 4, for Wilf equivalence and for growth rates. We establish bijective connections between pattern avoiding ascent sequences and various other combinatorial objects, in particular with set partitions. We also make a number of …
