Euler–Mahonian Statistics on Ordered Set Partitions

Abstract

An ordered partition with k blocks of $[n]:=\{1,2,\ldots, n\}$ is a sequence of k disjoint and nonempty subsets, called blocks, whose union is $[n]$. Clearly the number of such ordered partitions is $k!S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind. A statistic on ordered partitions of $[n]$ with k blocks is called an Euler–Mahonian statistic if its generating polynomial is $[k]_q!S_q(n,k)$, which is a natural q-analogue of $k!S(n,k)$. Motivated by Steingrímsson's conjectures dating back to 1997, we consider two different methods to produce Euler–Mahonian statistics on ordered set partitions: (a) we give a bijection between ordered partitions and weighted Motzkin paths by using a variant of Françon–Viennot's bijection to derive many Euler–Mahonian statistics by expanding the generating function of $[k]_q!S_q(n,k)$ as an explicit continued fraction; (b) we encode ordered partitions by walks in some digraphs and then derive new Euler–Mahonian statistics by computing their generating functions using the transfer-matrix method. In particular, we prove several conjectures of Steingrímsson.