Bijections on m-level Rook Placements

TL;DR

This paper develops explicit bijections for m-level rook placements on Ferrers boards, generalizing previous results and providing formulas for rook numbers, with implications for combinatorial enumeration.

Contribution

It introduces new bijections that generalize prior work on rook placements, including preservation of statistics and formulas for rook numbers.

Findings

Bijections between Ferrers boards with the same rook placements

A formula for rook numbers using elementary symmetric functions

Bijection between boards with the same hit numbers

Abstract

Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of m-level rook placements. The first generalizes a map by Foata and Sch\"utzenberger and our proof applies to any Ferrers board. This bijection also preserves the m-inversion number statistic of an m-level rook placement, defined by Briggs and Remmel. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards, but it yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne.