Euler-Mahonian Statistics On Ordered Set Partitions (II)

We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of \stein (Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. We also give a partition version of MacMahon's theorem on the equidistribution of the statistics inversion number and major index on words.Comment: 27 pages,8 figure

Euler-Mahonian statistics on ordered set partitions (II)

International audienceWe study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of Steingr\'{\i}msson (Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. We also give a partition version of MacMahon's theorem on the equidistribution of the statistics inversion number and major index on words

Statistics on ordered partitions of sets

We introduce several statistics on ordered partitions of sets, that is, set partitions where the blocks are permuted arbitrarily. The distribution of these statistics is closely related to the q-Stirling numbers of the second kind. Some of the statistics are generalizations of known statistics on set partitions, but others are entirely new. All the new ones are sums of two statistics, inspired by statistics on permutations, where one of the two statistics is based on a certain partial ordering of the blocks of a partition.Comment: Added a Prologue, as this paper is soon to be published in a journa

Statistics on Ordered Partitions of Sets and q-Stirling Numbers

An ordered partition of [n]:={1,2,..., n} is a sequence of its disjoint subsets whose union is [n]. The number of ordered partitions of [n] with k blocks is k!S(n,k), where S(n,k) is the Stirling number of second kind. In this paper we prove some refinements of this formula by showing that the generating function of some statistics on the set of ordered partitions of [n] with k blocks is a natural $q$-analogue of k!S(n,k). In particular, we prove several conjectures of Steingr\'{\i}msson. To this end, we construct a mapping from ordered partitions to walks in some digraphs and then, thanks to transfer-matrix method, we determine the corresponding generating functions by determinantal computations.Comment: 29 page

Generalized permutation patterns - a short survey

An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance—or the prescribed number of occurrences— of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns

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 surprising $q$-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This $q$-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain $q$-analogs, $(q,p)$-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Bj\"orner and Welker.Comment: Final version; to appear in Advances in Mathematics; 52 pages; this paper was originally part of the longer paper arXiv:0805.2416v1, which has been split into three paper