The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices.

In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials F m , n , 1 (x) and F m , n , 2 (x). Then, he defined A m (n , k) and B m (n , k) to be the polynomials satisfying F m , n , 1 (x) = ∑ k = 0 n A m (n , k) x n − k (x + 1) k and F m , n , 1 (x) = ∑ k = 0 n B m (n , k) x n − k (x + 1) k. In this paper, we give a combinatorial interpretation of the coefficients of A m + 1 (n , k) and prove a symmetry of the coefficients, i.e., [ m s ] A m + 1 (n , k) = [ m n − s ] A m + 1 (n , n − k). We give a combinatorial interpretation of B m + 1 (n , k) and prove that B m + 1 (n , n − 1) is a polynomial in m with non-negative integer coefficients. We also prove that if n ≥ 6 then all coefficients of B m + 1 (n , n − 2) except the coefficient of m n − 1 are non-negative integers. For all n , the coefficient of m n − 1 in B m + 1 (n , n − 2) is − (n − 1) , and when n ≤ 5 some other coefficients of B m + 1 (n , n − 2) are also negative. [ABSTRACT FROM AUTHOR]