Unimodality of a refinement of Lassalle's sequence

TL;DR

This paper proves that a refined sequence related to Lassalle's sequence, which counts certain uniquely sorted permutations, is unimodal, confirming a previous conjecture.

Contribution

The authors prove the conjecture that the refined sequences of Lassalle's sequence are unimodal, advancing understanding of permutation enumeration.

Findings

Sequences are symmetric in the first element.

Sequences are proven to be unimodal.

Supports previous conjecture on sequence properties.

Abstract

Defant, Engen, and Miller defined a refinement of Lassalle's sequence $A_{k+1}$ by considering uniquely sorted permutations of length $2k+1$ whose first element is $\ell$. They showed that each such sequence is symmetric in $\ell$ and conjectured that these sequences are unimodal. We prove that the sequences are unimodal.