Permutations with restricted positions or rook placements are common objects in enumerative combinatorics. Classically, for any subset or board of [n] x [n] we want to find the rook number r_i that counts the placements of i non-attacking rooks on B or the hit number h_i that counts permutation matrices of size n with i rooks in B. 

 

In 1986 Garsia and Remmel introduced polynomial q-analogues for rook and hit numbers for boards coming from Ferrers diagrams of partitions. Haglund (1998) related these q-analogues to matrices over finite fields. In 2011 with Lewis, Liu, Panova, Sam and Zhang we introduced a, not necessarily polynomial, q-analogue of rook numbers for all boards B as counting matrices over finite fields with support in B by rank. 

 

We continue this new q-rook theory by defining a q-analogue of hit numbers for all boards that coincides with the Garsia-Remmel q-hit numbers for Ferrers diagrams. We give a reciprocity relation of the matrix q-rook numbers using a "MacWilliams identity" of Delsarte from coding theory. We use this relation to show polynomiality of these new q-rook numbers for boards coming from inversions of permutations, settling a conjecture with Lewis and Klein (2014).

 

Joint work with Joel Lewis.