Quasi-Stirling Permutations on Multisets

A permutation $\pi$ of a multiset is said to be a {\em quasi-Stirling } permutation if there does not exist four indices $i<j<k<\ell$ such that $\pi_i=\pi_k$ and $\pi_j=\pi_{\ell}$. Define $$\overline{Q}_{\mathcal{M}}(t,u,v)=\sum_{\pi\in \overline{\mathcal{Q}}_{\mathcal{M}}}t^{des(\pi)}u^{asc(\pi)}v^{plat(\pi)},$$ where $\overline{\mathcal{Q}}_{\mathcal{M}}$ denotes the set of quasi-Stirling permutations on the multiset $\mathcal{M}$, and $asc(\pi)$ (resp. $des(\pi)$, $plat(\pi)$) denotes the number of ascents (resp. descents, plateaux) of $\pi$. Denote by $\mathcal{M}^{\sigma}$ the multiset $\{1^{\sigma_1}, 2^{\sigma_2}, \ldots, n^{\sigma_n}\}$, where $\sigma=(\sigma_1, \sigma_2, \ldots, \sigma_n)$ is an $n$-composition of $K$ for positive integers $K$ and $n$. In this paper, we show that $\overline{Q}_{\mathcal{M}^{\sigma}}(t,u,v)=\overline{Q}_{\mathcal{M}^{\tau}}(t,u,v)$ for any two $n$-compositions $\sigma$ and $\tau$ of $K$. This is accomplished by establishing an $(asc, des, plat)$-preserving bijection between $\overline{\mathcal{Q}}_{\mathcal{M}^{\sigma}}$ and $\overline{\mathcal{Q}}_{\mathcal{M}^{\tau}}$. As applications, we obtain generalizations of several results for quasi-Stirling permutations on $\mathcal{M}=\{1^k,2^k, \ldots, n^k\}$ obtained by Elizalde and solve an open problem posed by Elizalde.