Search

We derive functional equations for distributions of six classical statistics (ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, and right-to-left minima) on separable and irreducible separable permutations. The equations are used to find a third degree equation for joint distribution of ascents and descents on separable permutations that generalizes the respective known result for the descent distribution. Moreover, our general functional equations allow us to derive explicitly (joint) distribution of any subset of maxima and minima statistics on irreducible, reducible and all separable permutations. In particular, there are two equivalence classes of distributions of a pair of maxima or minima statistics. Finally, we present three unimodality conjectures about distributions of statistics on separable permutations., Comment: To appear in Discrete Applied Mathematics

A bijection between $(31245,32145,31254,32154)$-avoiding permutations and $(31425,32415,31524,32514)$-avoiding permutations is constructed, which preserves five classical set-valued statistics. Combining with two codings of permutations due respectively to Baril--Vajnovszki and Martinez--Savage proves an enumerative conjecture posed by Gao and Kitaev. Moreover, the generating function for the common counting sequence is proved to be algebraic., Comment: Final remarks are added: 28 pages, 6 figures

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 of equidistribution results involving various (multiset-valued) permutation statistics follow from these applications. As byproducts, seven apparently new Mahonian statistics present themselves; new interpretations of known Mahonian statistics are discovered as well. Finally, in our effort to show the interconnections between these Mahonian statistics, we are naturally led to a new link between the variant Yan-Zhou-Lin bijection and the Kreweras complement., Comment: 7 figures and 3 tables

The systematic study of inversion sequences avoiding triples of relations was initiated by Martinez and Savage. For a triple $(\rho_1,\rho_2,\rho_3)\in\{<,>,\leq,\geq,=,\neq,-\}^3$, they introduced $\I_n(\rho_1,\rho_2,\rho_3)$ as the set of inversion sequences $e=e_1e_2\cdots e_n$ of length $n$ such that there are no indices $1\leq i)$ and $\I_n(>,\neq,\geq)$ that preserves the distinct entries and further posed a symmetry conjecture of ascents on these two classes of restricted inversion sequences. Concerning Lin's symmetry conjecture, an algebraic proof using the kernel method was recently provided by Andrews and Chern, but a bijective proof still remains mysterious. The goal of this article is to establish bijectively both Lin's symmetry conjecture and the $\gamma$-positivity of the ascent polynomial on $\I_n(>,\neq,>)$. The latter result implies that the distribution of ascents on $\I_n(>,\neq,>)$ is symmetric and unimodal., Comment: 23 pages, 2 figures

Permutation statistics $\wnm$ and $\rlm$ are both arising from permutation tableaux. $\wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $\rlm$ is showed by Nadeau equally distributed with the number of $1$'s in the first row of a permutation tableau. In this paper, we investigate the joint distribution of $\wnm$ and $\rlm$. Statistic $(\rlm,\wnm,\rlmin,\des,(\underline{321}))$ is shown equally distributed with $(\rlm,\rlmin,\wnm,\des,(\underline{321}))$ on $S_n$. Then the generating function of $(\rlm,\wnm)$ follows. An involution is constructed to explain the symmetric property of the generating function. Also, we study the triple statistic $(\wnm,\rlm,\asc)$, which is shown to be equally distributed with $(\rlmax-1,\rlmin,\asc)$ as studied by Josuat-Verg$\grave{e}$s. The main method we adopt throughout the paper is constructing bijections based on a block decomposition of permutations., Comment: 19pages,1 figure

The notion of a 12-representable graph was introduced by Jones et al.. This notion generalizes the notions of the much studied permutation graphs and co-interval graphs. It is known that any 12-representable graph is a comparability graph, and also that a tree is 12-representable if and only if it is a double caterpillar. Moreover, Jones et al.\ initiated the study of 12-representability of induced subgraphs of a grid graph, and asked whether it is possible to characterize such graphs. This question in is meant to be about induced subgraphs of a grid graph that consist of squares, which we call square grid graphs. However, an induced subgraph in a grid graph does not have to contain entire squares, and we call such graphs line grid graphs. In this paper we answer the question of Jones et al.\ by providing a complete characterization of $12$-representable square grid graphs in terms of forbidden induced subgraphs. Moreover, we conjecture such a characterization for the line grid graphs and give a number of results towards solving this challenging conjecture. Our results are a major step in the direction of characterization of all 12-representable graphs since beyond our characterization, we also discuss relations between graph labelings and 12-representability, one of the key open questions in the area., Comment: To appear in Discussiones Mathematicae Graph Theory

The distribution of certain Mahonian statistic (called $\mathrm{BAST}$) introduced by Babson and Steingr\'{i}msson over the set of permutations that avoid vincular pattern $1\underline{32}$, is shown bijectively to match the distribution of major index over the same set. This new layer of equidistribution is then applied to give alternative interpretations of two related $q$-Stirling numbers of the second kind, studied by Carlitz and Gould. Moreover, extensions to an Euler-Mahonian statistic over ordered set partitions, and to statistics over ordered multiset partitions present themselves naturally. The latter of which is shown to be related to the recently proven Delta Conjecture. During the course, a refined relation between $\mathrm{BAST}$ and its reverse complement $\mathrm{STAT}$ is derived as well., Comment: 29 pages, a new extension to ordered multiset partitions added

Babson and Steingr\'{\i}msson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Recently, Amini investigated the equidistributions of these Mahonian statistics over sets of pattern avoiding permutations. Moreover, he posed several conjectures. In this paper, we construct a bijection from $S_n(213)$ to $S_n(231)$, which maps the statistic $(maj,stat)$ to the statistic $(stat,maj)$. This allows us to give solutions to some of Amini's conjectures., Comment: 10 pages

Babson and Steingr\'{\i}msson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $\pi$, let $des(\pi)$ denote the descent number of $\pi$ and $maj(\pi)$ denote the major index of $\pi$. Babson and Steingr\'{\i}msson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.

Permutation tableaux were introduced by Steingr\'{\i}msson and Williams. Corteel and Kim defined the sign of a permutation tableau in terms of the number of unrestricted columns. The sign-imbalance of permutation tableaux of length $n$ is the sum of signs over permutation tableaux of length $n$. They have btained a formula for the sign-imbalance of permutation tableaux of length $n$ by using generating functions and asked for a combinatorial proof. Moreover, they raised the question of finding a sign-imbalance formula for type $B$ permutation tableaux introduced by Lam and Williams. We define a statistic $\nwnm$ over permutations and show that the number of unrestricted columns over permutation tableaux of length $n$ is equally distributed with $\nwnm$ over permutations of length $n$. This leads to a combinatorial interpretation of the formula of Corteel and Kim. For type $B$ permutation tableaux, we define the sign of a type $B$ permutation tableau in term of the number of certain rows and columns. On the other hand, we construct a bijection between the type $B$ permutation tableaux of length $n$ and symmetric permutations of length $2n$ and we show that the statistic $\nwnm$ over symmetric permutations of length $2n$ is equally distributed with the number of certain rows and columns over type $B$ permutation tableaux of length $n$. Based on this correspondence and an involution on symmetric permutation of length $2n$, we obtain a sign-imbalance formula for type $B$ permutation tableaux., Comment: 18 pages

The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$. Chung, Claesson, Dukes and Graham obtained polynomials $P_k(x)$ that can be used to determine the number of permutations of $[n]$ with $d$ descents and maximum drop size not larger than $k$. Furthermore, Chung and Graham gave combinatorial interpretations of the coefficients of $Q_k(x)=x^k P_k(x)$ and $R_{n,k}(x)=Q_k(x)(1+x+\cdots+x^k)^{n-k}$, and raised the question of finding a bijective proof of the symmetry property of $R_{n,k}(x)$. In this paper, we establish a bijection $\varphi$ on $A_{n,k}$, where $A_{n,k}$ is the set of permutations of $[n]$ and maximum drop size not larger than $k$. The map $\varphi$ remains to be a bijection between certain subsets of $A_{n,k}$. %related to the symmetry property. This provides an answer to the question of Chung and Graham. The second result of this paper is a proof of a conjecture of Hyatt concerning the unimodality of polynomials in connection with the number of signed permutations of $[n]$ with $d$ type $B$ descents and the type $B$ maximum drop size not greater than $k$., Comment: 19 pages

An alternating permutation of length $n$ is a permutation $\pi=\pi_1 \pi_2 ... \pi_n$ such that $\pi_1 < \pi_2 > \pi_3 < \pi_4 > ...$. Let $A_n$ denote set of alternating permutations of ${1,2,..., n}$, and let $A_n(\sigma)$ be set of alternating permutations in $A_n$ that avoid a pattern $\sigma$. Recently, Lewis used generating trees to enumerate $A_{2n}(1234)$, $A_{2n}(2143)$ and $A_{2n+1}(2143)$, and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by B\'ona, Xu and Yan. In this paper, we prove the two relations $|A_{2n+1}(1243)|=|A_{2n+1}(2143)|$ and $|A_{2n}(4312)|=|A_{2n}(1234)|$ as conjectured by Lewis., Comment: 21 pages, 2 figures