Recently, Deutsch and Elizalde studied the largest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let A (n,k) (t) denote the total weight of partitions on [n + 1] = \1,2,..., n + 1\ with the largest singleton \k + 1\. In this paper, explicit formulas for A (n,k) (t) and many combinatorial identities involving A (n,k) (t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, [GRAPHICS] where D-k is the number of permutations of [k] with no fixed points.

Source : oai:HAL:hal-00990479v1

Volume: Vol. 13 no. 3

Section: Combinatorics

Published on: December 27, 2011

Submitted on: October 16, 2010

Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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