## 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

### Keywords:

inversion formula; recursion formula; Pascal identity