Fix-Mahonian Calculus III; a Quadruple Distribution

Mahonian STAT on rearrangement class of words

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…

Mahonian STAT on words

In 2000, Babson and Steingrímsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In 2010, Burstein refined some of these results by giving a bijection between permutations with a fixed value for the major index and those with the same value for STAT, where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson a…

Linear Turán numbers of acyclic quadruple systems

A linear $r$-uniform hypergraph is called acycilc if it can be constructed starting from one single edge then at each step adding a new edge that intersect the union of the vertices of the previous edges in at most one vertex. Recently, Gyárfás, Ruszinkó and Sárk\''{o}zy initiated the study of the linear Turán numbers of acyclic linear triple systems. In this paper, we extend their results to linear quadruple systems. Here, we concentrate on small trees, paths and matchings. For the case of small trees, we find that for a linear tree $T$, $ex^{lin}_{4}(n,T)$ relates to difficult problems on St…

Mesh patterns and the expansion of permutation statistics as sums of permutation patterns

Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= Σ_τλ_f(τ)τ$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(π,R)$ is an occurrence of the permutation pattern $π$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(π,R)$, we have $λ_p(τ) = (-1)^{|τ|-|π|}p^{\star}(τ)$ …

欧拉型多项式与匹配及匹配排列

Eulerian-type polynomials over matchings and matching permutations

本文研究了匹配与排列之间的联系，揭示了匹配中四元统计量与排列中相应统计量的对应关系，并通过引入匹配排列，扩展了五变量邻居多项式与二阶欧拉多项式的关系。

Eulerian quasisymmetric functions and poset topology

We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specialize…

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprisi…