An interpretation for Garsia and Remmel's q-hit numbers

摘要

We study Garsia and Remmel's q-analogue of the rook numbers and hit numbers associated with a Ferrers board. They define their q-rook numbers directly, and they give a three-part recurrence for the corresponding q-hit numbers. We find a new recurrence for these q-hit numbers, and we define a statistic xi on permutations that constitutes a direct combinatorial interpretation for them. The distribution of xi is invariant under column permutations of the board. For "staircase" boards, the distribution of xi is Eulerian-Mahonian. We use the statistic to prove a reciprocity theorem, and we give a new proof that the sequence of coefficents in the q-hit numbers is symmetric. (C) 1998 Academic Press. [References: 12]