Combinatorial and statistical applications of generalized Stirling numbers. (Q2781717)

The generalized Stirling numbers \(S(n,k;\alpha ,\beta ,\gamma )\) were defined by \textit{L. C. Hsu} and \textit{P. J.-S. Shiue} [Adv. Appl. Math. 20, 366--384 (1998; Zbl 0913.05006)]. The authors describe a combinatorial problem whose solution is given in terms of these numbers, with \(\alpha ,\beta ,\gamma\) being non-negative integers, \(\alpha \mid \beta\), \(\alpha \mid \gamma\). Namely, \(n\) distinct balls are distributed (one ball at a time) among \(k+1\) distinct cells, the first \(k\) of which have \(\beta\) distinct compartments in each cell, and the last cell has \(\gamma\) distinct compartments. The compartments in each cell are given cyclic ordered numbering. It is assumed that the capacity of each compartment is limited to one ball; each successive \(\alpha\) available compartments in a cell can only have the leading compartment getting the ball; the first \(k\) cells are non-empty.NEWLINENEWLINEThe number of ways to do the above task equals \(\beta^kk!S(n,k;\alpha ,\beta ,\gamma )\). Some modifications of this problem are also considered, as well as their statistical interpretations.