Permutations and combinations of colored multisets

Quaintance, Jocelyn ; Kwong, Harris

Journal of integer sequences, Tome 13 (2010) no. 2

Résumé

Given positive integers $ m$ and $ n$, let $ S_n^m$ be the $ m$-colored multiset $ \{1^m,2^m,\ldots,n^m\}$, where $ i^m$ denotes $ m$ copies of $ i$, each with a distinct color. This paper discusses two types of combinatorial identities associated with the permutations and combinations of $ S_n^m$. The first identity provides, for $ m\geq 2$, an $ (m-1)$-fold sum for $ {mn\choose n}$. The second type of identities can be expressed in terms of the Hermite polynomial, and counts color-symmetrical permutations of $ S_n^2$, which are permutations whose underlying uncolored permutations remain fixed after reflection and a permutation of the uncolored numbers.

Détail

Classification : 05A05, 05A15, 05A19, 33C45
Keywords: permutations, combinations, colored multisets, Hermite polynomials