Rook theory for perfect matchings

Abstract

In classical rook theory there is a fundamental relationship between the rook numbers and the hit numbers relative to any board. In that theory the k-th hit number of a board B can be interpreted as the number of permutations whose intersection with B is of size k. In the case of Ferrers boards there are q-analogues of the hit numbers and the rook numbers developed by A. M. Garsia and J. B. Remmel (1986, J. Combin. Theory, Ser. A 41, 246-275) M. Dworkin (1996, "Generalizations of Rook Polynomials," Ph. D. Thesis, Brandeis University"; 1998, J. Combin. Theory, Ser. A 81, 149-175) and J. Haglund (1998, Adv. Appl. Math. 20, 450-487). In this paper we develop a rook theory appropriate for shifted partitions, where hit numbers can be interpreted as the number of perfect matchings in the complete graph whose intersection with the board is of size k. We show there is also analogous q-theory for the rook and hit numbers for these shifted Ferrers boards. © 2001 Academic Press.