Schur-Concavity for Avoidance of Increasing Subsequences in Block-Ascending Permutations

For integers $a_1, \dots, a_n \ge 0$ and $k \ge 1$, let $\mathcal L_{k+2}(a_1,\dots, a_n)$ denote the set of permutations of $\{1, \dots, a_1+\dots+a_n\}$ whose descent set is contained in $\{a_1, a_1+a_2, \dots, a_1+\dots+a_{n-1}\}$, and which avoids the pattern $12\dots(k+2)$. We exhibit some bijections between such sets, most notably showing that $\# \mathcal L_{k+2} (a_1, \dots, a_n)$ is symmetric in the $a_i$ and is in fact Schur-concave. This generalizes a set of equivalences observed by Mei and Wang.