p-rook numbers and cycle counting in C p ≀ S n
- Authors
- Haglund, J.[Haglund, J.]; Remmel, J.B.[Remmel, J.B.]; Yoo, M.[Yoo, M.]
- Issue Date
- 2019
- Publisher
- Springer New York LLC
- Keywords
- Cycle-counting hit numbers; Cycle-counting rook numbers; Hit numbers; Rook numbers; Wreath product
- Citation
- Developments in Mathematics, v.58, pp.250 - 282
- Indexed
- SCOPUS
- Journal Title
- Developments in Mathematics
- Volume
- 58
- Start Page
- 250
- End Page
- 282
- URI
- https://scholarworks.bwise.kr/skku/handle/2021.sw.skku/15699
- DOI
- 10.1007/978-3-030-11102-1_12
- ISSN
- 1389-2177
- 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 Cycle-counting q-rook numbers were introduced by Ehrenborg, Haglund, and Readdy [unpublished] and cycle-counting q-hit numbers were introduced by Haglund [Adv.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 ≀ 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 ≀ 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 ≀ S n according to the signs of the cycles. © Springer Nature Switzerland AG 2019.
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