Rook-by-rook rook theory: Bijective proofs of rook and hit equivalences

摘要

Suppose mu and nu are integer partitions of n, and N > n. It is well known that the Ferrets boards associated to mu and nu are rook-equivalent iff the multisets vertical bar mu(i) + i : 1 <= i <= N vertical bar and vertical bar nu(i) + i : 1 <= i <= N vertical bar are equal. We use the Garsia-Milne involution principle to produce a bijective proof of this theorem in which non-attacking rook placements for mu are explicitly matched with corresponding placements for nu. One byproduct is a direct combinatorial proof that the matrix of Stirling numbers of the first kind is the inverse of the matrix of Stirling numbers of the second kind. We also prove q-analogues and p, q-analogues of these results. We also use the Garsia-Milne involution principle to show that for any two rook boards B and B', if B and B' are bijectively rook-equivalent, then B and B' are bijectively hit-equivalent.