Bijections on m-level Rook Placements

Kenneth Barrese,Bruce Sagan

doi:10.46298/dmtcs.2430

Abstract

Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two 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 Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also 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. Nous considérons les rangs d’un échiquier partagés en ensembles de $m$ rangs appelés les niveaux. Un $m$-placement des tours est un sous-ensemble des carrés du plateau tel qu’il n’y a pas deux carrés dans la même colonne ou dans le même niveau. Nous construisons deux bijections explicites entre des plateaux de Ferrers ayant les mêmes nombres de $m$-placements. La première est une généralisation d’une fonction de Foata et Schützenberger et notre démonstration est pour n’importe quels plateaux de Ferrers. La deuxième généralise une bijection de Loehr et Remmel. Cette construction marche seulement pour des plateaux particuliers, mais ça donne une formule pour le nombre de $m$-placements en terme des fonctions symétriques élémentaires. Enfin, nous généralisons un autre résultat de Loehr et Remmel donnant une bijection entre deux plateaux ayant les mêmes nombres de coups. Les deux dernières bijections utilisent le Principe des Involutions de Garsia et Milne.

Highlights

Rook theory is the study of the numbers rk(B), which count the number of ways to place k non-attacking rooks on a board B
The theorem of Foata and Schutzenberger was later proved as an elegant corollary to the Factorization Theorem of Goldman, Joichi, and White [GJW75], which gave a complete factorization of the rook polynomial of a Ferrers board over the intergers
Loehr and Remmel constructed a bijection between rook placements on rook equivalent Ferrers boards using the Garsia-Milne Involution Principle which implied the Factorization Theorem [LR09]

Summary

Introduction

Rook theory is the study of the numbers rk(B), which count the number of ways to place k non-attacking rooks on a board B. Note that this is neither the English nor the French style of writing Ferrers diagrams It is useful because we usually place rooks rooks on the board from left to right and enumerating the number of such placements is facilitated by our convention. For any non-negative integer k, an m-level rook placement of k rooks on B is a subset of cardinality k of the cells of B which contains no more than one cell from any given level or column of B.

Rook equivalence and bijections

A second bijection on m-level rook placements

A bijection for hit numbers

Open Problem