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Permutations and combinations of colored multisets. (English) Zbl 1283.05018
Summary: 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 $$\binom{mn}{n}$$. 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.
##### MSC:
 05A05 Permutations, words, matrices 05A15 Exact enumeration problems, generating functions 05A19 Combinatorial identities, bijective combinatorics 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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