On criteria for rook equivalence of Ferrers boards

TL;DR

This paper establishes a new criterion for rook equivalence of Ferrers boards by proving the converse of a previously introduced Wilf equivalence, and refines existing criteria using nested sequences of L's.

Contribution

It proves the equivalence between rook and Wilf equivalence for Ferrers boards and introduces a new criterion involving nested sequences of L's.

Findings

Rook equivalence implies Wilf equivalence.

Wilf equivalence implies rook equivalence.

New criterion for rook equivalence using nested sequences of L's.

Abstract

In [2] we introduced a new notion of Wilf equivalence of integer partitions and proved that rook equivalence implies Wilf equivalence. In the present paper we prove the converse and thereby establish a new criterion for rook equivalence. We also refine two of the standard criteria for rook equivalence and establish another new one involving what we call \emph{nested sequences of L's}.