p-Rook Numbers and Cycle Counting in Cp ≀ Sn

Cycle-counting rook numbers were introduced by Chung and Graham [7]. Cycle-counting q-rook numbers were introduced by Ehrenborg, Haglund, and Readdy [9] and cycle-counting q-hit numbers were introduced by Haglund [14]. Briggs and Remmel [4] introduced the theory of p-rook and p-hit numbers which is a rook theory model where the rook numbers correspond to partial permutations in Cp ≀ Sn, the wreath product of the cyclic group Cp and the symmetric group Sn, and the hit numbers correspond to signed permutations in Cp ≀ Sn. In this paper, we extend the cycle-counting q-rook numbers and cycle-counting q-hit numbers to the Briggs-Remmel model. In such a setting, we define a multivariable version of the cycle-counting q-rook numbers and cycle-counting q-hit numbers where we keep track of cycles of permutations and partial permutations of Cp ≀ Sn according to the signs of the cycles.