Abstract
A permutation \(\pi \) of a multiset is said to be a quasi-Stirling permutation if there do not exist four indices \(i<j<k<\ell \) such that \(\pi _i=\pi _k\), \(\pi _j=\pi _{\ell }\) and \(\pi _i\ne \pi _j\). Define
where \(\overline{\mathcal {Q}}_{\mathcal {M}}\) denotes the set of quasi-Stirling permutations of the multiset \(\mathcal {M}\), and \(\mathrm{asc}(\pi )\) (resp. \(\mathrm{des}(\pi )\), \(\mathrm{plat}(\pi )\)) 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 \((\mathrm{asc}, \mathrm{des}, \mathrm{plat})\)-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.
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The author would like to thank the referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (12071440).
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Yan, S.H.F., Yang, L., Huang, Y. et al. Statistics on quasi-Stirling permutations of multisets. J Algebr Comb 55, 1265–1277 (2022). https://doi.org/10.1007/s10801-021-01093-z
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DOI: https://doi.org/10.1007/s10801-021-01093-z