NDLI: EulerMahonian Statistics on Ordered Set Partitions

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ishikawa, Masao Zeng, Jiang Kasraoui, Anisse
Copyright Year	2008
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 EulerMahonian 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 Steingrmsson's conjectures dating back to 1997, we consider two different methods to produce EulerMahonian statistics on ordered set partitions: (a) we give a bijection between ordered partitions and weighted Motzkin paths by using a variant of FranonViennot's bijection to derive many EulerMahonian 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 EulerMahonian statistics by computing their generating functions using the transfer-matrix method. In particular, we prove several conjectures of Steingrmsson.
Starting Page	1105
Ending Page	1137
Page Count	33
File Format	PDF
ISSN	08954801
DOI	10.1137/060672340
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	3
Volume Number	22
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-07-02
Access Restriction	Subscribed
Subject Keyword	transfer-matrix method Exact enumeration problems, generating functions q-Stirling numbers of the second kind ordered partitions EulerMahonian statistics $q$-calculus and related topics Partitions of sets
Content Type	Text
Resource Type	Article
Subject	Mathematics

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