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Partition statistics on permutations

✍ Scribed by Paul H. Edelman; Rodica Simion; Dennis White

Publisher: Elsevier Science
Year: 1992
Tongue: English
Weight: 336 KB
Volume: 99
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(92)90366-n

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✦ Synopsis

Edelman, P.H., R. Simion and D. White, Partition statistics on permutations, Discrete Mathematics 99 (1992) 63-68.

We describe some properties of a new statistic on permutations. This statistic is closely related to a well-known statistic on set partitions.

In [7] four statistics on set partitions were described, each having the q-Stirling numbers [2] as their distribution generating function. In this paper we will show an interesting relationship between one of these statistics, Is, defined below, and permutations. Specifically, we compute the distribution polynomial c qW,

where P is, in turn, the weak and strong order on the symmetric group. In each case the polynomial has an interesting factorization. Our formulae (l), (3), and (4) are instances of the following phenomenon: A poset P and a combinatorial

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