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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.
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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