Quasi-Stirling polynomials on multisets

Abstract

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