Cycles and perfect matchings

Fan Chung and Ron Graham (J. Combin. Theory Ser. B 65 (1995), 273-290) introduced the cover polynomial for a directed graph and showed that it was connected with classical rook theory. M. Dworkin (J. Combin. Theory Ser. B 71 (1997), 17-53) showed that the cover polynomial naturally factors for directed graphs associated with Ferrers boards. The authors (Adv. Appl. Math. 27 (2001), 438-481) developed a rook theory for shifted Ferrers boards where the analogue of a rook placement is replaced by a partial perfect matching of K2n, the complete graph on 2n vertices. In this paper, we show that an analogue of Dworkin’s result holds for shifted Ferrers boards in this setting. We also show how cycle-counting matching numbers are connected to cycle-counting “hit numbers” (which involve perfect matchings of K2n). Introduction Let B2n be the board pictured in Fig. 1. Let (i, j) denote the square in the i-th row and j-th column of B2n, so B2n = {(i, j) : 1 ≤ i < j ≤ 2n}. Let K2n denote the complete graph on vertices {1, 2, . . . , 2n}. A perfect matching of K2n is a set of n edges of K2n where no two edges have a vertex in common. Given a perfect matching m of K2n, we let pm = {(i, j) : i < j and {i, j} ∈ m}. For example, if m = {{1, 4}, {2, 7}, {3, 5}, {6, 8}} is a perfect matching of K8, then pm is pictured in Fig. 2. For a given board B ⊆ B2n, we say that a subset p ⊆ B is a rook placement of B if there is a perfect matching m of K2n such that p ⊆ pm. We let Mk(B) denote the set of all k element perfect matchings of B and we call mk(B) = |Mk(B)| the k-th rook number of B. We let Fk,2n(B) = {pm : |pm ∩B| = k and m is a perfect matching of K2n}. We call fk,2n(B) = |Fk,2n(B)| the k-th hit number of B. Haglund and Remmel [HR] proved the following relationship between the hit numbers and the rook numbers of a board B ⊆ B2n. 1991 Mathematics Subject Classification. 05A05.