Greene--Kleitman invariants for Sulzgruber insertion

TL;DR

This paper establishes a new connection between Sulzgruber's rim hook insertion and the Robinson-Schensted-Knuth correspondence, revealing that Sulzgruber's map can be described using Greene-Kleitman invariants, thus linking two important combinatorial bijections.

Contribution

The paper introduces a diagonal RSK description of Sulzgruber's insertion and demonstrates its equivalence to Hillman-Grassl, enabling expression of Sulzgruber's map via Greene-Kleitman invariants.

Findings

Sulzgruber's insertion is equivalent to a form of RSK called diagonal RSK.

Sulzgruber's map can be expressed using Greene-Kleitman invariants.

The work unifies two bijections in the theory of reverse plane partitions.

Abstract

R. Sulzgruber's rim hook insertion and the Hillman-Grassl correspondence are two distinct bijections between the reverse plane partitions of a fixed partition shape and multisets of rim-hooks of the same partition shape. It is known that Hillman-Grassl may be equivalently defined using the Robinson-Schensted-Knuth correspondence, and we show the analogous result for Sulzgruber's insertion. We refer to our description of Sulzgruber's insertion as diagonal RSK. As a consequence of this equivalence, we show that Sulzgruber's map from multisets of rim hooks to reverse plane partitions can be expressed in terms of Greene-Kleitman invariants.