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Journal Articles Discrete Mathematics and Theoretical Computer Science Year : 2011

The largest singletons in weighted set partitions and its applications

Abstract

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.
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Dates and versions

hal-00990479 , version 1 (13-05-2014)

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Yidong Sun, Yanjie Xu. The largest singletons in weighted set partitions and its applications. Discrete Mathematics and Theoretical Computer Science, 2011, Vol. 13 no. 3 (3), pp.75--86. ⟨10.46298/dmtcs.535⟩. ⟨hal-00990479⟩

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