Euler-mahonian Statistics on Ordered Partitions and Steingrímsson’s Conjecture — a Survey
نویسندگان
چکیده
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]. In this article, we consider Steingŕımsson’s conjectures about Euler-Mahonian statistics on ordered partitions dated back to 1997. We encode ordered partitions by walks in some digraphs and then derive their generating functions using the transfer-matrix method. In particular, we prove half of Steingŕımsson’s conjectures by the computation of the resulting determinants. This article is a very short version of our paper: “Statistics on Ordered Partitions of Sets and q-Stirling Numbers” (arxiv:math.CO/0605390), announcing and surveying some of the results in it.
منابع مشابه
Euler-mahonian Statistics on Ordered Partitions
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 Euler-Mahonian statistics if its generating polynomial is [k]q!Sq(n...
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