M-Rook numbers and a generalization of a formula of Frobenius to [formula omitted

Abstract

In ordinary rook theory, rook placements are associated to permutations of the symmetric group S n . We provide a generalization of this theory in which “ m-rook placements” are related to elements of C m ≀ S n , where C m is the cyclic group of order m. Within this model, we define and interpret combinatorially a p , q -analogue of the m-rook numbers. We also define a p , q -analogue of the m-hit numbers and show that the coefficients of these polynomials in p and q are nonnegative integers for m-Ferrers boards. Finally, we define statistics des m ( σ ) , maj m ( σ ) , and comaj m ( σ ) as analogues of the ordinary descent, major, and comajor index statistics and prove a generalization of a formula of Frobenius that relates these statistics to generalized p , q -Stirling numbers of the second kind.