Degenerate Whitney Numbers of First and Second Kind of Dowling Lattices

Abstract

Dowling constructed Dowling lattice Qn(G), for any finite set with n elements and any finite multiplicative group G of order m, which is a finite geometric lattice. He also defined the Whitney numbers of the first and second kinds for any finite geometric lattice. These numbers for the Dowling lattice Qn(G) are the Whitney numbers of the first kind Vm(n,k) and those of the second kind Wm(n,k), which are given by Stirling number-like relations. In this paper, by `degenerating' such relations we introduce the degenerate Whitney numbers of the first kind and those of the second kind and investigate, among other things, generating functions, recurrence relations and various explicit expressions for them. As further generalizations of the degenerate Whitney numbers of both kinds, we also consider the degenerate r-Whitney numbers of both kinds.