Refined restricted inversion sequences

Recently, the study of patterns in inversion sequences was initiated by Corteel-Martinez-Savage-Weselcouch and Mansour-Shattuck independently. Motivated by their works and a double Eulerian equidistribution due to Foata (1977), we investigate several classical statistics on restricted inversion sequences that are either known or conjectured to be enumerated by {\em Catalan}, {\em Large Schr\"oder}, {\em Baxter} and {\em Euler} numbers. One of the two highlights of our results is a fascinating bijection between $000$-avoiding inversion sequences and Simsun permutations, which together with Foata's V- and S-codes, provide a proof of a restriced double Eulerian equdistribution. The other one is a refinement of a conjecture due to Martinez and Savage that the cardinality of \I_n(\geq,\geq,>) is the $n$-th Baxter number, which is proved via the so-called {\em obstinate kernel method} developed by Bousquet-M\'elou.Comment: 25 pages, 6 figures. This is the full version of the extended abstract that appears in FPSAC'17 Londo

Further equidistribution of set-valued statistics on permutations

We construct bijections to show that two pairs of sextuple set-valued statistics of permutations are equidistributed on symmetric groups. This extends a recent result of Sokal and the second author valid for integer-valued statistics as well as a previous result of Foata and Han for bivariable set-valued statistics.Comment: 14 page

A combinatorial bijection on di-sk trees

A di-sk tree is a rooted binary tree whose nodes are labeled by $\oplus$ or $\ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees which proves that the two quadruples (\LMAX,\DESB,\iar,\comp) and (\LMAX,\DESB,\comp,\iar) have the same distribution over separable permutations. Here for a permutation $\pi$, \LMAX(\pi) is the set of values of the left-to-right maxima of $\pi$ and \DESB(\pi) is the set of descent bottoms of $\pi$, while \comp(\pi) and \iar(\pi) are respectively the number of components of $\pi$ and the length of initial ascending run of $\pi$. Interestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides (up to the classical {\em Knuth--Richards bijection}) an alternative approach to a result of Rubey (2016) that asserts the two triples (\LMAX,\iar,\comp) and (\LMAX,\comp,\iar) are equidistributed on $321$-avoiding permutations. Rubey's result is a symmetric extension of an equidistribution due to Adin--Bagno--Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive. Some equidistribution results for various statistics concerning tree traversal are presented in the end.Comment: 20 pages, 7 figure

Refined Wilf-equivalences by Comtet statistics

We launch a systematic study of the refined Wilf-equivalences by the statistics $\mathsf{comp}$ and $\mathsf{iar}$, where $\mathsf{comp}(\pi)$ and $\mathsf{iar}(\pi)$ are the number of components and the length of the initial ascending run of a permutation $\pi$, respectively. As Comtet was the first one to consider the statistic $\mathsf{comp}$ in his book {\em Analyse combinatoire}, any statistic equidistributed with $\mathsf{comp}$ over a class of permutations is called by us a {\em Comtet statistic} over such class. This work is motivated by a triple equidistribution result of Rubey on $321$-avoiding permutations, and a recent result of the first and third authors that $\mathsf{iar}$ is a Comtet statistic over separable permutations. Some highlights of our results are: (1) Bijective proofs of the symmetry of the double Comtet distribution $(\mathsf{comp},\mathsf{iar})$ over several Catalan and Schr\"oder classes, preserving the values of the left-to-right maxima. (2) A complete classification of $\mathsf{comp}$- and $\mathsf{iar}$-Wilf-equivalences for length $3$ patterns and pairs of length $3$ patterns. Calculations of the $(\mathsf{des},\mathsf{iar},\mathsf{comp})$ generating functions over these pattern avoiding classes and separable permutations. (3) A further refinement by the Comtet statistic $\mathsf{iar}$, of Wang's recent descent-double descent-Wilf equivalence between separable permutations and $(2413,4213)$-avoiding permutations.Comment: 39 pages, 2 tables, 2 figures. Comments are welcom

Plateaux on generalized Stirling permutations and partial γ\gamma-positivity

We prove that the enumerative polynomials of generalized Stirling permutations by the statistics of plateaux, descents and ascents are partial $\gamma$-positive. Specialization of our result to the Jacobi-Stirling permutations confirms a recent partial $\gamma$-positivity conjecture due to Ma, Yeh and the second named author. Our partial $\gamma$-positivity expansion, as well as a combinatorial interpretation for the corresponding $\gamma$-coefficients, are obtained via the machine of context-free grammars and a group action on generalized Stirling permutations. Besides, we also provide an alternative approach to the partial $\gamma$-positivity from the stability of certain multivariate polynomials.Comment: 11 pages, 1 figur

Consecutive Patterns in Inversion Sequences

An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of patterns in inversion sequences, focusing on the enumeration of those that avoid classical patterns of length 3. We initiate an analogous systematic study of consecutive patterns in inversion sequences, namely patterns whose entries are required to occur in adjacent positions. We enumerate inversion sequences that avoid consecutive patterns of length 3, and generalize some results to patterns of arbitrary length. Additionally, we study the notion of Wilf equivalence of consecutive patterns in inversion sequences, as well as generalizations of this notion analogous to those studied for permutation patterns. We classify patterns of length up to 4 according to the corresponding Wilf equivalence relations.Comment: Final version to appear in DMTC

A new decomposition of ascent sequences and Euler--Stirling statistics

As shown by Bousquet-M\'elou--Claesson--Dukes--Kitaev (2010), ascent sequences can be used to encode $({\bf2+2})$-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series $$\sum_{m=1}^{\infty}\prod_{i=1}^m (1-(1-t)^i).$$ In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: (i) a calculation of the Euler--Stirling distribution on ascent sequences, including the numbers of ascents (\asc), repeated entries (\rep), zeros (\zero) and maximal entries ($\max$). In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of \zero and $\max$. (ii) a far-reaching generalization of the generating function formula for (\asc,\zero) due to Jel\'inek. This is accomplished via a bijective proof of the quadruple equidistribution of (\asc,\rep,\zero,\max) and (\rep,\asc,\rmin,\zero), where \rmin denotes the right-to-left minima statistic of ascent sequences. (iii) an extension of a conjecture posed by Levande, which asserts that the pair (\asc,\zero) on ascent sequences has the same distribution as the pair (\rep,\max) on $({\bf2-1})$-avoiding inversion sequences. This is achieved via a decomposition of $({\bf2-1})$-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples (\asc,\rep,\zero,\max) and (\rep,\asc,\max,\zero) are equidistributed on ascent sequences.Comment: 25 pages, to appear in Journal of Combinatorial Theory, Series

Inversion sequences avoiding pairs of patterns

The enumeration of inversion sequences avoiding a single pattern was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck independently. Their work has sparked various investigations of generalized patterns in inversion sequences, including patterns of relation triples by Martinez and Savage, consecutive patterns by Auli and Elizalde, and vincular patterns by Lin and Yan. In this paper, we carried out the systematic study of inversion sequences avoiding two patterns of length $3$. Our enumerative results establish further connections to the OEIS sequences and some classical combinatorial objects, such as restricted permutations, weighted ordered trees and set partitions. Since patterns of relation triples are some special multiple patterns of length $3$, our results complement the work by Martinez and Savage. In particular, one of their conjectures regarding the enumeration of $(021,120)$-avoiding inversion sequences is solved

Consecutive patterns in inversion sequences II: avoiding patterns of relations

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 relations $R_{1},R_2\in\{\leq,\geq,,=,\neq\}$, we study inversion sequences $e$ with no subindex $i$ such that $e_{i}R_{1}e_{i+1}R_{2}e_{i+2}$. By enumerating such inversion sequences according to their length, we obtain well-known quantities such as Catalan numbers, Fibonacci numbers and central polynomial numbers, relating inversion sequences to other combinatorial structures. We also classify consecutive patterns of relations into Wilf equivalence classes, according to the number of inversion sequences avoiding them, and into more restrictive classes that consider the positions of the occurrences of the patterns. As a byproduct of our techniques, we obtain a simple bijective proof of a result of Baxter--Shattuck and Kasraoui about Wilf-equivalence of vincular patterns, and we prove a conjecture of Martinez and Savage, as well as related enumeration formulas for inversion sequences satisfying certain unimodality conditions.Comment: 32 papes, 9 figure