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. Thus we first study generating functions for the number of even or odd involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern $τ$ on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. Next, we consider other statistics on 132-avoiding involutions by counting an occurrences of some generalized patterns, related to the enumeration according to the number of rises.
We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions.
We study generating functions for the number of involutions, even involutions, and odd involutions in $S_n$ subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern $τ$ or contain $τ$ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials of the second kind.
Several authors have examined connections among restricted permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for involutions which avoid 3412. Our results include a recursive procedure for computing the generating function for involutions which avoid 3412 and any set of additional patterns. We use our results to give enumerations and generating functions for involutions which avoid 3412 and various sets of additional patterns. In many cases we express these generating functions in terms of Chebyshev polynomials of…
In this paper we prove a refined major-balance identity on the $321$-avoiding involutions of length $n$, respecting the leading element of permutations. The proof is based on a sign-reversing involution on the lattice paths within a $\lfloor\frac{n}{2}\rfloor\times\lceil\frac{n}{2}\rceil$ rectangle. Moreover, we prove affirmatively a question about refined major-balance identity on the $123$-avoiding involutions, respecting the number of descents.
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…
Let $A_k$ be the set of permutations in the symmetric group $S_k$ with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns $A_k$. We present a bijection between symmetric Schroder paths of length $2n$ and involutions of length $n+1$ avoiding $\mathcal{A}_4$. Statistics such as the number of right-to-left maxima and fixed points of the involution correspond to the number of steps in the symmetric Schroder path of a particular type. For each $k> 2$ we determine the generating function for the number of involutions avoiding the subsequences in $A_k$, acco…
Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations for the moments of Laguerre polynomials via Flajolet's combinatorial theory of continued fractions. In this paper, we first introduce a reflection-like involution on restricted Laguerre histories. Then, we demonstrate its power by composing this involution with three bijections due to Fran\ccon-Viennot, Foata-Zeilberger, and Yan-Zhou-Lin, respectively. A host …
Define $I_n^k(α)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $α\in S_i$, for some $i \geq 2$, and define $I_n^k(\emptyset;α)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which contain the pattern $α\in S_i$, for some $i \geq 2$, exactly once. Let $i_n^k(α)$ be the number of elements in $I_n^k(α)$ and let $i_n^k(\emptyset;α)$ be the number of elements in $I_n^k(\emptyset;α)$. We investigate $I_n^k(α)$ and $I_n^k(\emptyset;α)$ for all $α\in S_3$. In particular, we show that $i_n^k(132)=i_n^k(213)=i_n^k(3…
Centrosymmetric involutions in the symmetric group S_{2n} are permutations πsuch that π=π^{-1} and π(i)+π(2n+1-i)=2n+1 for all i, and they are in bijection with involutions of the hyperoctahedral group. We describe the distribution of some natural descent statistics on 321-avoiding centrosymmetric involutions, including the number of descents in the first half of the involution, and the sum of the positions of these descents. Our results are based on two new bijections, one between centrosymmetric involutions in S_{2n} and subsets of {1,...,n}, and another one showing that certain statistics o…
Let $W$ be a finite Coxeter group and $X$ a subset of $W$. The length polynomial $L_{W,X}(t)$ is defined by $L_{W,X}(t) = \sum_{x \in X} t^{\ell(x)}$, where $\ell$ is the length function on $W$. In this article we derive expressions for the length polynomial where $X$ is any conjugacy class of involutions, or the set of all involutions, in any finite Coxeter group $W$. In particular, these results correct errors in the paper "Permutation statistics on involutions", W.M.B. Dukes., European J. Combin. 28 (2007), 186--198. for the involution length polynomials of Coxeter groups of type $B_n$ and …
We establish a combinatorial connection between the sequence $(i_{n,k})$ counting the involutions on $n$ letters with $k$ descents and the sequence $(a_{n,k})$ enumerating the semistandard Young tableaux on $n$ cells with $k$ symbols. This allows us to show that the sequences $(i_{n,k})$ are not log-concave for some values of $n$, hence answering a conjecture due to F. Brenti.
We use Naito-Sagaki's work [S. Naito & D. Sagaki, J. Algebra 245 (2001) 395--412, J. Algebra 251 (2002) 461--474] on Lakshmibai-Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type $A_{2\ell}^{(2)}$ and of type $D_{\ell+1}^{(2)}$. We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijectio…
In this paper we look at polynomials arising from statistics on the classes of involutions, $I_n$, and involutions with no fixed points, $J_n$, in the symmetric group. Our results are motivated by F. Brenti's conjecture which states that the Eulerian distribution of $I_n$ is log-concave. Symmetry of the generating functions is shown for the statistics $des,maj$ and the joint distribution $(des,maj)$. We show that $exc$ is log-concave on $I_n$, $inv$ is log-concave on $J_n$ and $des$ is partially unimodal on both $I_n$ and $J_n$. We also give recurrences and explicit forms for the generating fu…
We present an involution on set partitions that interchanges two statistics related to relative size of block entries and use it to establish an equidistribution on objects counted by the Bessel numbers.
We describe an involution on Tamari intervals and m-Tamari intervals. This involution switches two sets of statistics known as the "rises" and the "contacts" and so proves an open conjecture from Préville-Ratelle on intervals of the m-Tamari lattice.
In [Decompositions and statistics for β(1,0)-trees and nonseparable permutations, Advances Appl. Math. 42 (2009) 313--328] we introduced an involution, h, on β(1,0)-trees. We neglected, however, to prove that h indeed is an involution. In this note we provide the missing proof. We also refine an equidistribution result given in the same paper.
Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $σ\in\mathfrak{S}_n$ contains a pattern $π\in\mathfrak{S}_k$ if $σ$ has a subsequence of length $k$ whose letters are in the same relative order as $π$. This paper is a comprehensive study of the same two statistics, number of inversions and major index, over involutions $\mathcal{I}_n=\{σ\in\mathfrak{S}_n:σ^2=\text{id}\}$ that avoid one or more length three patterns. The equalities between the generating functions are con…
We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin paths is demonstrated.
We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns t_1...t_{m-2}m(m-1) and t_1...t_{m-2}(m-1)m in a permutation, respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As special case we derive a one-to-one correspondence between permutations which avoid each of the patterns t_1...t_{m-2}m(m-1) in S_m and such ones which avoid each of the patterns t_1...t_{m-2}(m-1)m. For m=3, this correspondence coincides with the bijection given by S…
In 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on $R(w)$, the rearrangement class o…
We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al.~ and the bijection between $(321,…
In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect matchings.
A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equi-distributed with the standard one. This concept is then applied to construct explicit cyclic descent extensions on involutions, standard Young tableaux and Motzkin paths. Schur-positivity of the associated quasisymmetric functions follows.
We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.
There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\tilde{S}_n$ on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are length-preserving in an appropriate sense, from the set of involutions in $\tilde{S}_n$ to the set of $\mathbb{N}$-weighted matchings in the $n$-element cycle graph. As an application, we compute a formula for the bivariate generating function counting the involutions in $\tilde S_n$ by length and absolute length. The 0-Hecke monoid of $\tilde{S}_n$ also acts on…
A symmetry of $(t,q)$-Eulerian numbers of type $B$ is combinatorially proved by defining an involution preserving many important statistics on the set of permutation tableaux of type $B$. This involution also proves a symmetry of the generating polynomial $\hat{D}_{n, k}(p,q,r)$ of number of crossings and alignments, and hence $q$-Eulerian numbers of type $A$ defined by L. Williams. By considering a restriction of our bijection, we were led to define a new statistic on the permutations of type $D$ and $(t,q)$-Eulerian numbers of type $D$, which is proved to have a nice symmetry as well. We con…
In this undergraduate thesis, we expand on the study of statistics on restricted growth functions avoiding patterns initiated by Campbell, et. al. Restricted growth functions are of interest because they are in bijection with set partitions. We examine the case when a restricted growth function contains a pattern exactly $k$ times, where $k=0$ corresponds to pattern avoidance. We prove results for several statistic and pattern combinations in the case when $k=1$ and define a new type of Wilf equivalence for these patterns. We also examine a poset related to $M(n)$, the poset of integer partiti…
We study the total number of occurrences of several vincular (also called generalized) patterns and other statistics, such as the major index and the Denert statistic, on permutations avoiding a pattern of length 3, extending results of Bona (2010, 2012) and Homberger (2012). In particular, for 2-3-1-avoiding permutations, we find the total number of occurrences of any vincular pattern of length 3. In some cases the answer is given by simple expressions involving binomial coefficients. The tools we use are bijections with Dyck paths, generating functions, and block decompositions of permutatio…
A restricted growth function (RGF) of length n is a sequence w = w_1 w_2 ... w_n of positive integers such that w_1 = 1 and w_i is at most 1 + max{w_1,..., w_{i-1}} for i at least 2. RGFs are of interest because they are in natural bijection with set partitions of {1, 2, ..., n}. RGF w avoids RGF v if there is no subword of w which standardizes to v. We study the generating functions sum_{w in R_n(v)} q^{st(w)} where R_n(v) is the set of RGFs of length n which avoid v and st(w) is any of the four fundamental statistics on RGFs defined by Wachs and White. These generating functions exhibit inte…
The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice through a stack (where the letters on the stack have to be in increasing order) was conjectured by West, and later proved by Zeilberger. Goulden and West found a bijection from such permutations to nonseparable planar maps, and later, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called beta(1,0)-trees. Using…
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…
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…
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…
We prove that the Mahonian-Stirling pairs of permutation statistics $(\sor, \cyc)$ and $(\inv, \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters. Moreover, we define a sor…
We present a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. This gives a simple combinatorial proof of recent results of Robertson, Saracino and Zeilberger, and the first author. We also show that our bijection preserves additional statistics, which extends the previous results.
A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of interest to this work. We first characterize the restricted growth functions of several avoidance classes based on partitions of size four, enumerate these avoidance classes, and consider the distribution of the Wachs and White statistics across these avoidance classes. We also investigate the equidistribution of statistics between avoidance classes based on mu…
We consider the question of computing the distribution of a permutation statistics over restricted permutations via enumeration schemes. The restricted permutations are those avoiding sets of vincular patterns (which include both classical and consecutive patterns), and the statistics are described in the number of copies of certain vincular patterns such as the descent statistic and major index. An enumeration scheme is a polynomial-time algorithm (specifically, a system of recurrence relations) to compute the number of permutations avoiding a given set of vincular patterns. Enumeration schem…
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 …
In this thesis, we introduced and carried out a combinatorial study of permutations that avoid one or two patterns of length 3 according to the statistic number of crossings. For this purpose, we manipulated a bijection of Elizalde and Pak and constructed other bijections that preserve the number of crossings. As results, we found, throughout these bijections, various relationships on the distributions of the number of crossings on restricted permutations as well as combinatorial interpretations in terms of the number of crossings on permutations with forbidden patterns of some well known tria…
Let $st=\{st_1,\ldots,st_k\}$ be a set of $k$ statistics on permutations with $k\geq 1$. We say that two given subset of permutations $T$ and $T'$ are $st$-Wilf-equivalent if the joint distributions of all statistics in $st$ over the sets of $T$-avoiding permutations $S_n(T)$ and $T'$-avoiding permutations $S_n(T')$ are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single patterns in $S_3$, where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection $Θ:S_n(321)\rightarrow S_…
A Refinement of the Fixed--Pixed Points Equidistribution on restricted Permutations
College of Mathematics and Statistics, Northwest Normal University（西北师范大学数学与统计学院）；Universite Lyon 1, UMR 5208 du CNRS, Institut Camille Jordan（里昂第一大学，CNRS第5208联合研究实验室，卡米尔·乔丹研究所）
Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture, but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted per…
We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths (i.e. Motzkin paths with all level steps on the $x$-axis) with a given number of peaks. We show how the bijection maps statistics of consecutive patterns of faro words into linear combinations of other pattern statistics on paths. Then, we deduce enumerative results by providing multivariate gen…
