Euler-Mahonian statistics on ordered set partitions

Masao Ishikawa, Anisse Kasraoui, Jiang Zeng

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


An ordered partition with k blocks of [n] := {1,2,...,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!Sq(n, k), which is a natural q-analogue of k!S(n, k). Motivated by Steingrfmsson'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!Sq(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 Steingrfmsson.

Original languageEnglish
Pages (from-to)1105-1137
Number of pages33
JournalSIAM Journal on Discrete Mathematics
Issue number3
Publication statusPublished - Jan 1 2008
Externally publishedYes


  • Euler-Mahonian statistics
  • Ordered partitions
  • Q-stirling numbers of the second kind
  • Transfer-matrix method

ASJC Scopus subject areas

  • Mathematics(all)


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