Stack-Sorting, Set Partitions, and Lassalle's Sequence

TL;DR

This paper establishes a bijection between valid hook configurations and weighted set partitions, providing new combinatorial interpretations of Lassalle's sequence and exploring properties of permutations related to stack-sorting.

Contribution

It introduces a novel bijection linking combinatorial objects and set partitions, and offers new interpretations and recurrence relations involving Lassalle's sequence.

Findings

Identified a bijection between valid hook configurations and weighted set partitions.

Provided three new combinatorial interpretations of Lassalle's sequence.

Proved symmetry and conjectured log-concavity of permutation sequences related to stack-sorting.

Abstract

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new combinatorial interpretations of Lassalle's sequence. One of these interpretations involves permutations that have exactly one preimage under the (West) stack-sorting map. We prove that the sequences obtained by counting these permutations according to their first entries are symmetric, and we conjecture that they are log-concave. We also obtain new recurrence relations involving Lassalle's sequence and the sequence that enumerates valid hook configurations. We end with several suggestions for future work.