A polynomial generalization of some associated sequences related to set partitions

Abstract

In this paper, we consider a common polynomial generalization, denoted by $w_m(n,k)=w_m^{a,b,c,d}(n,k)$, of several types of associated sequences. When $a=0$ and $b=1$, one gets a generalized associated Lah sequence, while if $c=0$, $d=1$, one gets a polynomial array that enumerates a restricted class of weighted ordered partitions of size n having k blocks. The particular cases when $a=d=0$ and $b=c=1$ or when $a=c=0$ and $b=d=1$ correspond to the associated Stirling numbers of the first and second kind, respectively. We derive several combinatorial properties satisfied by $w_m(n,k)$ and consider further the case $a=0$, $b=1$. Our results not only generalize prior formulas found for the associated Stirling and Lah numbers but also yield some apparently new identities for these sequences. Finally, explicit exponential generating function formulas for $w_m(n,k)$ are derived in the cases when $a=0$ or $b=0$.