Limit Theorems for Associated Whitney Numbers of Dowling Lattices

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 \( \max_k U_m{(n,k)} \), \( \max_k V_m{(n,k)} \) and their modes are given.