Quasi-Stirling Permutations on Multisets

Abstract

A permutation $π$ 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 $π_i=π_k$ and $π_j=π_{\ell}$. Define $$ \overline{Q}_{\mathcal{M}}(t,u,v)=\sum_{π\in \overline{\mathcal{Q}}_{\mathcal{M}}}t^{des(π)}u^{asc(π)}v^{plat(π)},$$ where $\overline{\mathcal{Q}}_{\mathcal{M}}$ denotes the set of quasi-Stirling permutations on the multiset $\mathcal{M}$, and $asc(π)$ (resp. $des(π)$, $plat(π)$) denotes the number of ascents (resp. descents, plateaux) of $π$. Denote by $\mathcal{M}^σ$ the multiset $\{1^{σ_1}, 2^{σ_2}, \ldots, n^{σ_n}\}$, where $σ=(σ_1, σ_2, \ldots, σ_n)$ is an $n$-composition of $K$ for positive integers $K$ and $n$. In this paper, we show that $\overline{Q}_{\mathcal{M}^σ}(t,u,v)=\overline{Q}_{\mathcal{M}^τ}(t,u,v)$ for any two $n$-compositions $σ$ and $τ$ of $K$. This is accomplished by establishing an $(asc, des, plat)$-preserving bijection between $\overline{\mathcal{Q}}_{\mathcal{M}^σ}$ and $\overline{\mathcal{Q}}_{\mathcal{M}^τ}$. 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.