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A common generalization of binomial coefficients, Stirling numbers and Gaussian coefficients. (English) Zbl 0558.05003

The generalization of the title is: For finite sets \(A_ 0,A_ 1,A_ 2,..\). with \(a_ i=| A_ i|\) and nonnegative integers n and k denote by \(S^ n_ k(a_ 0,a_ 1,a_ 2,...)\) the number of words \(w=(w_ 0,...,w_{n-1})\) such that (1) w contains k labels, say at positions \(i_ 0,...,i_{k-1}\), (2) all entries in w before position \(i_ 0\) belong to \(A_ 0\), all entries in w between \(i_ j\) and \(i_{j+1}\), where \(j=0,...,k-2\), belong to \(A_{j+1}\), all entries after position \(i_{k-1}\) belong to \(A_ k\). The author gives explicit characterization of the numbers \(S^ n_ k\) and proves several recursion and inversion formulae involving them.
Reviewer: St.Porubsky

MSC:

05A15 Exact enumeration problems, generating functions
05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics
11A07 Congruences; primitive roots; residue systems
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