r-Whitney numbers of Dowling lattices

Let G be a finite group of order m ≥ 1 . A Dowling lattice Q n ( G ) is the geometric lattice of rank n over G . In this paper, we define the r -Whitney numbers of the first and second kind over Q n ( G ) , respectively. This concept is a common generalization of the Whitney numbers and the r -Stirling numbers of both kinds. We give their combinatorial interpretations over the Dowling lattice and we obtain various new algebraic identities. In addition, we develop the r -Whitney–Lah numbers and the r -Dowling polynomials associated with the Dowling lattice.