2021/11/10 by Shishuo Fu, Yanlin Li, Fu, Shishuo +1
Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2111.05758
openalex publication_date 2021/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a general multiset M=\1m1,2m2,…,nmn\, where i appears mi times, a multipermutation π of M is called \em 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 \em 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.