Sophie Huczynska, Vincent Vatter
We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length n in a permutation class is either at least as large as the nth Fibonacci number or is eventually polynomial.
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We give enumerations of various families of restricted permutations involving the Fibonacci numbers or k-generalized Fibonacci numbers.
A permutation $π\in S_n$ is said to {\it avoid} a permutation $σ\in S_k$ whenever $π$ contains no subsequence with all of the same pairwise comparisons as $σ$. For any set $R$ of permutations, we write $S_n(R)$ to denote the set of permutations in $S_n$ which avoid every permutation in $R$. In 1985 Simion and Schmidt showed that $|S_n(132, 213, 123)|$ is equal to the Fibonacci number $F_{n+1}$. In this paper we generalize this result in several ways. We first use a result of Mansour to show that for any permutation $τ$ in a certain infinite family of permutations, $|S_n(132, 213, τ)|$ is given…
In a recent paper, Goyt and Sagan studied distributions of certain set partition statistics over pattern restricted sets of set partitions that were counted by the Fibonacci numbers. Their study produced a class of $q$-Fibonacci numbers, which they related to $q$-Fibonacci numbers studied by Carlitz and Cigler. In this paper we will study the distributions of some Mahonian statistics over pattern restricted sets of permutations. We will give bijective proofs connecting some of our $q$-Fibonacci numbers to those of Carlitz, Cigler, Goyt and Sagan. We encode these permutations as words and use a…
It is well known that permutations avoiding any 3-length pattern are enumerated by the Catalan numbers. If the three patterns 123, 132 and 213 are avoided at the same time we obtain a class of permutations enumerated by the Fibonacci numbers. We start from these permutations and make one or two forbidden patterns disappear by suitably "generalizing" them. In such a way we find several classes of permutations enumerated by integer sequences which lay between the Fibonacci and Catalan numbers. For each class, we provide the generating function according to the length of the permutations. Moreove…
We prove that $|Av_n(231,312,1432)|$, $|Av_n(312,321,1342)|$ $|Av_n(231,312,4321,21543)|$, and $ |Av_n(321,231,4123,21534)|$, are all equal to $F_{n+1} - 1$ where $F_n$ is the $n$-th Fibonacci number using the convention $F_0 = F_1 = 1$ and $Av_n(S)$ is the set of all permutations of length $n$ that avoid all of the patterns in the set $S$. To do this, we characterize the structures of the permutations in these sets in terms of Fibonacci permutations. Then, we further quantify the structures using statistics such as inversion number and a statistic that measures the length of Fibonacci subsequ…
Odd-indexed Fibonacci numbers via pattern-avoiding permutations
本文研究了若干组合问题，其计数结果为奇数索引斐波那契数，包括特定类型的Dyck路径、块喷泉、有向列凸多边形和无交叉无嵌套的集合划分。通过构造到避免模式排列的双射，并推导生成函数以跟踪排列层面的统计量。
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…
In recent years there has been much interest in certain subcubes of hypercubes, namely Fibonacci cubes and Lucas cubes (and their generalized versions). In this article we consider online routing of linear permutations on these cubes. The model of routing we use regards edges as bi-directional, and we do not allow queues of length greater than one. Messages start out at different vertices, and in movements synchronized with a clock, move to an adjacent vertex or remain where they are, so that at the next stage there is still exactly one message per vertex. This is the routing model we defined …
The Collatz map is defined for a positive even integer as half that integer, and for a positive odd integer as that integer threefold, plus one. The Collatz conjecture states that when the map is iterated the number one is eventually reached. We study permutations that arise as sequences from this iteration. We show that permutations of this type of length up to 14 are enumerated by the Fibonacci numbers. Beyond that excess permutations appear. We will explain the appearance of these excess permutations and give an upper bound on the exact enumeration.
Fibonacci-like partitions and their associated piecewise-defined permutations
本文引入 Fibonacci 型划分，细化 Wythoff 划分，研究其上的分段置换，并应用于 Shapovalov 和 Venkatachala 的序列，推广得到猜想产生无限正交阵列的序列族。
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases is given by a sequence related to Fibonacci numbers. We also show that there is exactly one permutation whose inverse peak number is zero among all permutations with any fixed descent composition, and we give a few elementary consequences of this fact. Our proofs involve generating functions, symmetric functions, regular expressions, and monomino-domino til…
We study $q$-analogues of $k$-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most $k$. The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics. We use these $q$-analogues to produce $q$-analogues of identities involving $k$-Fibonacci numbers. This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations. In this paper we give general $q$-analogues of $k$-Fibonacci identities for arbitrary weights that depe…
Let $S_n$ denote the symmetric group on $\{1,2,\ldots,n\}$. For two permutations $u, v\in S_n$ such that $u\leq v$ in the Bruhat order, let $R_{u,v}(q)$ and $\R_{u,v}(q)$ denote the Kazhdan-Lusztig $R$-polynomial and $\R$-polynomial, respectively. Let $v_n=34\cdots n\, 12$, and let $σ$ be a permutation such that $σ\leq v_n$. We obtain a formula for the $\R$-polynomials $\R_{σ,v_n}(q)$ in terms of the $q$-Fibonacci numbers depending on a parameter determined by the reduced expression of $σ$. When $σ$ is the identity $e$, this reduces to a formula obtained by Pagliacci. In another direction, we …
The diameter of a graph is the maximum distance between pairs of vertices in the graph. A pair of vertices whose distance is equal to its diameter are called diametrically opposite vertices. The collection of shortest paths between diametrically opposite vertices are referred as diametral paths. In this work, we enumerate the number of diametral paths for Fibonacci cubes, Lucas cubes and Alternate Lucas cubes. We present bijective proofs that show that these numbers are related to alternating permutations and are enumerated by Euler numbers.
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.
We consider uniform random permutations in classes having a finite combinatorial specification for the substitution decomposition. These classes include (but are not limited to) all permutation classes with a finite number of simple permutations. Our goal is to study their limiting behavior in the sense of permutons. The limit depends on the structure of the specification restricted to families with the largest growth rate. When it is strongly connected, two cases occur. If the associated system of equations is linear, the limiting permuton is a deterministic $X$-shape. Otherwise, the limiting…
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.
A permutation is simsun if for all k, the subword of the one-line notation consisting of the k smallest entries does not have three consecutive decreasing elements. Simsun permutations were introduced by Simion and Sundaram, who showed that they are counted by the Euler numbers. In this paper we enumerate simsun permutations avoiding a pattern or a set of patterns of length 3. The results involve Motkzin, Fibonacci, and secondary structure numbers. The techniques in the proofs include generating functions, bijections into lattice paths and generating trees.
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.
Permutation Pattern Matching (PPM) is the problem of deciding for a given pair of permutations P and T whether the pattern P is contained in the text T. Bose, Buss and Lubiw showed that PPM is NP-complete. In view of this result, it is natural to ask how the situation changes when we restrict the pattern P to a fixed permutation class C; this is known as the C-Pattern PPM problem. Grid classes are special kind of permutation classes, consisting of permutations admitting a grid-like decomposition into simpler building blocks. Of particular interest are the so-called monotone grid classes, in wh…
The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, Schröder, Pell, even index Fibonacci numbers and the central binomial coefficients. Consequently, this provides Gray…
Enumeration of pattern-avoiding alternating sign matrices: An asymptotic dichotomy
本文对经典避免单个置换模式的交错符号矩阵的计数渐近行为给出完整分类，证明十一个特殊模式有指数上界，其余有超指数下界，并精确枚举同时避免2143和3412且含3个负一的矩阵。
We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, subject to the constraint that a third element of a suitable type be in a suitable position. For various instances of the problem, we compute the number of equivalence classes, determine how man…
The Fibonacci-Fubini and Lucas-Fubini numbers
本文通过将有序贝尔数中的划分块按斐波那契方式排列，引入斐波那契-富比尼数，并研究其生成函数、显式公式、组合解释及算术三角形，同时揭示其与斯特林数的联系。
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.
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…
Lehmer Codes and the Reverse-Complement Mapping from (32-1)-Avoiding Permutations to (3-21)-Avoiding Permutations
该研究以避免特定vincular模式的排列为对象，借助莱默码将其转化为加权偏序集，证明其极大元由斐波那契数计数，且经典逆补映射在极大元间构成双射，揭示了偏序集结构的对称性。
Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the sense of pattern avoidance. One of the celebrated notions in pattern theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if #Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage proposed studying a q-analogue of this concept defined as follows. Suppose st:S->N is a permutation statistic where N represents the nonnegativ…
A Fishburn permutation is a permutation which avoids the bivincular pattern $(231, \{1\}, \{1\})$, while an ascent sequence is a sequence of nonnegative integers in which each entry is less than or equal to one more than the number of ascents to its left. Fishburn permutations and ascent sequences are linked by a bijection $g$ of Bousquet-Mélou, Claesson, Dukes, and Kitaev. We write $F_n(σ_1,\ldots,σ_k)$ to denote the set of Fishburn permutations of length $n$ which avoid each of $σ_1,\ldots,σ_k$ and we write $A_n(α_1,\ldots,α_k)$ to denote the set of ascent sequences which avoid each of $α_1,…
Despite the fact that the field of pattern avoiding permutations has been skyrocketing over the last two decades, there are very few exhaustive generating algorithms for such classes of permutations. In this paper we introduce the notions of regular and colored regular set of forbidden patterns, which are particular cases of right-justified sets of forbidden patterns. We show the (colored) regularity of several sets of forbidden patterns (some of them involving variable length patterns) and we derive a general framework for the efficient generation of permutations avoiding them. The obtained g…
We study sorting by queues that can rearrange their content by applying permutations from a predefined set. These new sorting devices are called shuffle queues and we investigate those of them corresponding to sets of permutations defining some well-known shuffling methods. If $\mathbb{Q}_Σ$ is the shuffle queue corresponding to the shuffling method $Σ$, then we find a number of surprising results related to two natural variations of shuffle queues denoted by $\mathbb{Q}_Σ^{\prime}$ and $\mathbb{Q}_Σ^{\textsf{pop}}$. These require the entire content of the device to be unloaded after a permuta…
Denote by $A_n$ the set of square $(0,1)$ matrices of order $n$. The set $A_n$, $n\le8$, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular $(0,1)$ matrices of order 8 is 10160459763342013440. Let $D_n$, $S_n$ denote the set of absolute determinant values and Smith normal forms of matrices from $A_n$. Denote by $a_n$ the smallest integer not in $D_n$. The sets $\mathcal{D}_9$ and $\mathcal{S}_9$ are obtained; especially, $a_9=103$. The lower bounds for $a_n$, $10\le n\le 19$, (exceeding the …
Pattern Avoidance for Fibonacci Sequences using $k$-Regular Words
本文通过证明k-正则词避免特定模式的数量与斐波那契序列的关系，揭示了模式回避与组合结构之间的联系。
Patterns in Inversion Sequences I
本文首次提出反演序列中的模式概念，并枚举避免长度为三的单词的反演序列，其结果与斐波那契数、贝尔数、施罗德数和欧拉上下数等著名数列相关联。
Desarrangements revisited: statistics and pattern avoidance
本文重新研究首次上升位置为偶数的错位排列，给出按多种排列统计量计数的生成函数公式，并完整枚举回避长度3模式的错位排列，为 Catalan、Fine、Jacobsthal 和 Fibonacci 数提供了新解释。
A Fibonacci-Lucas based statistical model and several other related models are studied. The canonical and grand canonical partition functions for these models are developed.Partition structure such as the distribution of sizes as in a cluster distribution is explored.Ensemble averaging over all partitions leads to a scale invariant power law behavior at a particular critical like point. The canonical ensemble of the Fibonacci-Lucas case involves the Gegenbauer polynomial.The model has a hyperbolic power law behavior, a feature linked to the golden mean ratio of two adjacent Fibonacci numbers a…
In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of convex polyominoes, the k-convex polyominoes for which the enumeration according to the semi-perimeter is known only for k=1,2. We obtain, from a recursive decomposition, the generating function of the class of k-convex parallelogram polyominoes, which turns out to be rational. Noting that this generating function can be expressed in terms of the Fibonacci polyno…
Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. Several well-known graph classes have this property, including chordal graphs, permutation graphs, circular-arc graphs, and circle graphs. We perform a systematic study of the question which classes of graphs defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators. We focus on set…
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