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Permutations and Combinations of Colored
Multisets Jocelyn Quaintance
Department of Mathematics
West Virginia University
Morgantown, WV 26506
USA
mailto:jquainta@math.wvu.edujquainta@math.wvu.edu

Harris Kwong
SUNY Fredonia
Department of Mathematical Sciences
State University of New York at Fredonia
Fredonia, NY 14063
USA
mailto:kwong@fredonia.edukwong@fredonia.edu

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Abstract:

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.