m-Rook numbers and a generalization of a formula of Frobenius to Cm≀Sn

AbstractIn ordinary rook theory, rook placements are associated to permutations of the symmetric group Sn. We provide a generalization of this theory in which “m-rook placements” are related to elements of Cm≀Sn, where Cm 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 desm(σ), majm(σ), and comajm(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