### Abstract

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}!S_{q}(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}!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 Steingrfmsson.

Original language | English |
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Pages (from-to) | 1105-1137 |

Number of pages | 33 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2008 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*22*(3), 1105-1137. https://doi.org/10.1137/060672340