P-Rook Numbers and Cycle Counting in $$C_p \wr S_n$$ C p ≀ S n

Abstract

Cycle-counting rook numbers were introduced by Chung and Graham [J. Combin. Theory Ser. B 65 (1995), 273–290]. Cycle-counting q-rook numbers were introduced by Ehrenborg, Haglund, and Readdy [unpublished] and cycle-counting q-hit numbers were introduced by Haglund [Adv. Appl. Math. 17 (1996), 408–459]. Briggs and Remmel [J. Combin. Theory Ser. A 113 (2006), 1138–1171] 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 \(C_p \wr S_n\), the wreath product of the cyclic group \(C_p\) and the symmetric group \(S_n\), and the hit numbers correspond to permutations in \(C_p \wr S_n\). 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 \(C_p \wr S_n\) according to the signs of the cycles.