Unimodality of a refinement of Lassalle's sequence

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.