Quasi-Stirling polynomials on multisets

A permutation π of a multiset is said to be quasi-Stirling if there do not exist four indices i<j<k<ℓ such that πi=πk, πj=πℓ and πi≠πj. For M, denote by Q‾M the set permutations M. The polynomial on M defined Q‾M(t)=∑π∈Q‾Mtdes(π), where des(π) denotes number descents π. By employing generating function arguments, Elizalde derived an elegant identity involving polynomials {12,22,…,n2}, in analogy Stirling polynomials. In this paper, we derive Q‾M(t) for any which generalization Eulerian Elizalde's {12,22,…,n2}. We provide combinatorial proof terms certain ordered labeled trees. Specializing M={12,22,…,n2} implies answer problem posed Elizalde. As application, our enables us show has only real roots coefficients are unimodal log-concave Brenti's result multisets.