# 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 Steingrimsson'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 Francon-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 Steingrimsson.
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