An Interpretation for Garsia and Remmel'sq-Hit Numbers

Abstract

We study Garsia and Remmel'sq-analogue of the rook numbers and hit numbers associated with a Ferrers board. They define theirq-rook numbers directly, and they give a three-part recurrence for the correspondingq-hit numbers. We find a new recurrence for theseq-hit numbers, and we define a statisticξon permutations that constitutes a direct combinatorial interpretation for them. The distribution ofξis invariant under column permutations of the board. For “staircase” boards, the distribution ofξis Eulerian–Mahonian. We use the statistic to prove a reciprocity theorem, and we give a new proof that the sequence of coefficents in theq-hit numbers is symmetric.