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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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