Inserting rim-hooks into reverse plane partitions

TL;DR

This paper introduces a new algorithm for inserting rim-hooks into reverse plane partitions, establishing a bijection that provides a combinatorial proof of a hook-length formula for their generating function.

Contribution

It presents a novel insertion algorithm for rim-hooks, creating a bijection that connects reverse plane partitions with rim-hooks and relates to existing combinatorial correspondences.

Findings

Provides a new bijective proof of the hook-length formula

Establishes a connection between rim-hook insertion and known correspondences

Shows the equivalence of the new bijection with existing maps

Abstract

A new algorithm for inserting rim-hooks into reverse plane partitions is presented. The insertion is used to define a bijection between reverse plane partitions of a fixed shape and multi-sets of rim-hooks. In turn this yields a bijective proof of the fact that the generating function for reverse plane partitions of a fixed shape, which was first obtained by R. Stanley, factors into a product featuring the hook-lengths of this shape. Our bijection turns out to be equivalent to a map defined by I. Pak by different means, and can be related to the Hillman-Grassl correspondence and the Robinson-Schensted-Knuth correspondence.