We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern-classes $I_n(000, 021), I_n(100, 021)$, $I_n(110, 021), I_n(102, 021)$, $I_n(100,012)$, $I_n(011,201)$, $I_n(011,210)$ and $I_n(120,210)$. Then we use the kernel method, obtain generating functions of each class, and find enumerating formulas. Lin and Yan studied the classification of the Wilf-equivalences for inversion sequences avoiding pairs of length-three patterns and showed that there are 48 Wilf classes among 78 pairs. In this paper, we solve six open cases for such pattern classes.
Recently, Bill Chen, together with his disciples Alvin Dai and Robin Zhou, discovered, and very elegantly proved, an algebraic equation satisfied by the generating function enumerating 123-avoiding words with two occurrences of each of 1, ..., n. Inspired by this result, we present an algorithm for finding such an algebraic equation for the ordinary generating function enumerating 123-avoiding words with exactly r occurrences of each of 1, ... n for any positive integer r, thereby proving that they are algebraic and not merely D-finite (a fact that is promised by WZ theory). Our algorithm cons…
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.
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.
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…
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…
Enumerating Pattern-Avoiding Involutions using Combinatorial Exploration
School of Computer Science and Mathematics, Keele University（基尔大学计算机科学与数学学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）；Department of Mathematical and Statistical Sciences, Marquette University（马凯特大学数学与统计科学系）
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.
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…
The method we have applied in "A. Bernini, L. Ferrari, R. Pinzani, Enumerating permutations avoiding three Babson-Steingrimsson patterns, Ann. Comb. 9 (2005), 137--162" to count pattern avoiding permutations is adapted to words. As an application, we enumerate several classes of words simultaneously avoiding two generalized patterns of length 3.
A \Def{composition} of a positive integer $n$ is a $k$-tuple $(ł_1, ł_2, \dots, ł_k) \in \Z_{> 0}^k$ such that $n = ł_1 + ł_2 + \dots + ł_k$. Our goal is to enumerate those compositions whose parts $ł_1, ł_2, \dots, ł_k$ avoid a fixed arithmetic sequence. When this sequence is given by the even integers (i.e., all parts of the compositions must be odd), it is well known that the number of compositions is given by the Fibonacci sequence. A much more recent theorem says that when the parts are required to avoid all multiples of a given integer $k$, the resulting compositions are counted by a seq…
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…
We present a new approach to the problem of enumerating permutations of length n that avoid a fixed consecutive pattern of length m. We use this idea to give explicit upper and lower bounds on the number of permutations avoiding a pattern of length m. As a corollary, we obtain a simple proof of the CMP conjecture, regarding the most avoided pattern, recently shown by Elizalde. Finally, we also show that most of the patterns behave similar to the least avoided one.
The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Lin, and Kim and Lin. The strength of our approach is its robustness: we enumerate four families $F_1 \subset F_2 \subset F_3 \subset F_4$ of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family $F_i…
We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-Mélou. W…
Bivariate generating functions for various subsets of the class of permutations containing no descending sequence of length three or more are determined. The notion of absolute indecomposability of a permutation is introduced, and used in enumerating permutations which have a block structure avoiding 321 and whose blocks also have such structure (recursively). Generalizations of these results are discussed.
A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1<w_2>w_3<w_4>\cdots$ (when the word is up-down) or $w_1>w_2<w_3>w_4<\cdots$ (when the word is down-up). In this paper, we initiate the study of (pattern-avoiding) alternating words. We enumerate up-down (equivalently, down-up) words via finding a bijection with order ideals of a certain poset. Further, we show that the number of 123-avoiding up-down words of even length is given by the Narayana numbers, which is also the case, shown by us bijectively, with 132-avoiding up-down words of even length. We also give formulas for enumerating a…
Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of l…
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…
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 …
Chen and collaborators give a recursively defined bijection from 021-avoiding ascent sequences to 021-avoiding (aka 132-avoiding) permutations. Here we give an algorithmic bijection from 021-avoiding ascent sequences to Dyck paths. Our bijection does not appear to be closely related to the Chen bijection but, like the Chen bijection, it preserves several interesting statistics.
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of partitions avoiding an arbitrary set of "patterns".
Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of desc…
In this paper we introduce {\em weak ascent sequences}, a class of number sequences that properly contains ascent sequences. We show how these sequences uniquely encode each of the following objects: permutations avoiding a particular length-4 bivincular pattern; upper-triangular binary matrices that satisfy a column-adjacency rule; factorial posets that are weakly (3+1)-free. We also show how weak ascent sequences are related to a class of pattern avoiding inversion sequences that has been a topic of recent research by Auli and Elizalde. Finally, we consider the problem of enumerating these n…
We consider the problem of enumerating permutations in the symmetric group on $n$ elements which avoid a given set of consecutive pattern $S$, and in particular computing asymptotics as $n$ tends to infinity. We develop a general method which solves this enumeration problem using the spectral theory of integral operators on $L^{2}([0,1]^{m})$, where the patterns in $S$ has length $m+1$. Kre\uın and Rutman's generalization of the Perron--Frobenius theory of non-negative matrices plays a central role. Our methods give detailed asymptotic expansions and allow for explicit computation of leading t…
Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the classical (non-consecutive) case, and later by Auli--Elizalde in the consecutive case, where the entries of a pattern are required to occur in adjacent positions. In this paper we continue this investigation by considering {\em consecutive patterns of relations}, in analogy to the work of Martinez--Savage in the classical case. Specifically, given two binary re…
Pattern avoidance in alternating sign rectangles I: Extended avoidance
University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学物理学院）；Institute for Mathematics, Physics and Mechanics（数学、物理与力学研究所）；KTH Royal Institute of Technology（瑞典皇家理工学院）
Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern $σ\in\mathcal{S}_k$ if in each tree, each sequence of labels along the shortest path from the root to a vertex does not contain a subsequence with the same relative order as $σ$. For each permutation $σ\in\mathcal{S}_{k-2}$, we construct a bijection between $n$-vertex forests avoiding $(σ)(k-1)k:=σ(1)\cdotsσ(k-2)(k-1)k$ and $n$-vertex forests avoiding $(σ)k(k-1):=σ(1)\cdotsσ(k-2)k(k-1)$, giving a common generalization of results of West on permutations and Anders--Archer on forests. We further define…
We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function. We also complete the classification of consecutive …
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy all…
