Rook and Wilf equivalence of integer partitions

TL;DR

This paper introduces a new Wilf equivalence for integer partitions, demonstrating that rook equivalence implies Wilf equivalence and that refined notions of both coincide, advancing understanding of their relationship.

Contribution

It establishes a new notion of Wilf equivalence for integer partitions and proves the equivalence relations between rook and Wilf equivalence under refined conditions.

Findings

Rook equivalence implies Wilf equivalence.

Refined rook and Wilf equivalence notions coincide.

Wilf equivalence implies rook equivalence (from previous work).

Abstract

The subjects of rook equivalence and Wilf equivalence have both attracted considerable attention over the last half-century. In this paper we introduce a new notion of Wilf equivalence for integer partitions, and, using this notion, we prove that rook equivalence implies Wilf equivalence. We also prove that if we refine the notions of rook and Wilf equivalence in a natural way, then these two notions coincide. In [6] we prove that Wilf equivalence implies rook equivalence.