Statistics on wreath products and generalized binomial-Stirling numbers (Q2382351)

Fix four integers \(a,d,r,\ell\) and let \(g(n,k)=g_{a,d,r,\ell}(n,k) \) be the numbers defined by the recurrence \(g(0,0)=1\) and \(g(n,k)=(an+dk-r)\cdot g(n-1,k)+\ell \cdot g(n-1,k-1) \) and \(g(n,k)=0\) if \(k<0\) or \(n>k\). These numbers are common generalizations of the binomial coefficients and the Stirling numbers. For example \(g_{1,0,1,1}(n,k) \) are the signless Stirling numbers of the first kind, \(g_{0,1,0,1}(n,k) \) are the Stirling numbers of the second kind, and \(g_{0,0,-1,1}(n,k) \) are the binomial coefficients. Several sequences introduces in the paper fall into this general recurrence relation. The paper studies canonical presentations in wreath products and introduce statistics counting the number of ``long'' and ``short'' factors in these presentations. These numbers essentially count the number of certain right to left minima in colored permutations. It is shown that enumeration of elements in wreath products with respect to these (and to these and descent) statistics have nice recurrence formulas of binomial-Stirling type. In particular a wreath product extension of the Stirling numbers of the first and second kind is given and a MacMahon-type equidistribution theorem is shown.