Rooted quasi-Stirling permutations of general multisets
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Given a general multiset ℳ={1^m_1,2^m_2,…,n^m_n}, where i appears m_i times, a multipermutation π of ℳ is called quasi-Stirling, if it contains no subword of the form abab with a≠ b. We designate exactly one entry of π, say k∈ℳ, which is not the leftmost entry among all entries with the same value, by underlining it in π, and we refer to the pair (π,k) as a quasi-Stirling multipermutation of ℳ rooted at k. By introducing certain vertex and edge labeled trees, we give a new bijective proof of an identity due to Yan, Yang, Huang and Zhu, which links the enumerator of rooted quasi-Stirling multipermutations by the numbers of ascents, descents, and plateaus, with the exponential generating function of the bivariate Eulerian polynomials. This identity can be viewed as a natural extension of Elizalde's result on k-quasi-Stirling permutations, and our bijective approach to proving it enables us to: (1) prove bijectively a Carlitz type identity involving quasi-Stirling polynomials on multisets that was first obtained by Yan and Zhu; (2) confirm a recent partial γ-positivity conjecture due to Lin, Ma and Zhang, and find a combinatorial interpretation of the γ-coefficients in terms of two new statistics defined on quasi-Stirling multipermutations called sibling descents and double sibling descents.
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