Limit Theorems for Associated Whitney Numbers of Dowling Lattices

Abstract

The Whitney number $W_m(n,k)$ of the rank-$n$ Dowling lattice $Q_n(G)$ based on the group $G$ having order $m$ is the number of elements in $Q_n(G)$ of co-rank $k$. The associated numbers $U_m(n,k)=k!W_m(n,k)$ and $V_m(n,k)=k!m^k{W}_m(n,k)$ were studied by M. Benoumhani [\emph{Adv. in Appl. Math}. 19 (1997), no. 1, 106-116] where a generating function was derived using algebraic techniques and logconcavity was shown for $\{U_m(n,k)\}$ and for $\{V_m(n,k)\}$. We give a central limit theorem and a local limit theorem on $\mathbb{R}$ for $\{U_m(n,k)\}$ and for $\{V_m(n,k)\}$. In addition, asymptotic formulas for $\underset{k}{max}{U}_m(n,k)$, $\underset{k}{max}{V}_m(n,k)$ and their modes are given.