p-Rook Numbers and Cycle Counting in Cp o Sn

Cycle-counting rook numbers were introduced by Chung and Graham [8]. Cycle-counting q-rook numbers were introduced by Ehrenborg, Haglund, and Readdy [10] and cycle-counting q-hit numbers were introduced by Haglund [14]. Briggs and Remmel [5] 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 oSn, the wreath product of the cyclic group Cp and the symmetric group Sn, and the hit numbers correspond to permutations in Cp oSn. In this paper, we extend the cyclecounting q-rook numbers and cycle-counting q-hit numbers to the BriggsRemmel 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 o Sn according to the signs of the cycles.