Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.6

Permutations and Combinations of Colored Multisets

Jocelyn Quaintance
Department of Mathematics
West Virginia University
Morgantown, WV 26506

Harris Kwong
SUNY Fredonia
Department of Mathematical Sciences
State University of New York at Fredonia
Fredonia, NY 14063


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.

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(Concerned with sequence A047974.)

Received June 22 2009; revised versions received July 3 2009; November 9 2009; December 21 2009; February 18 2010. Published in Journal of Integer Sequences, February 18 2010.

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