On the [formula omitted]-positivity of multiset Eulerian polynomials

Abstract

Let An(p)(t) (resp. Bn(p)(t)) be the descent polynomial on permutations of the multiset {1p,2p,…,np} (resp. {1p,2p,…,np,n+1}). The γ-positivity of An(p)(t) was known but to give a combinatorial interpretation for the corresponding γ-coefficients still remains open. In this paper we• manage to find a combinatorial interpretation for the γ-coefficients of An(p)(t) via the model of weakly increasing trees;• and prove that Bn(p)(t) has bi-γ-positivity expansion Bn(p)(t)=an(t)+tbn(t), where bn(t)=(p−1)An(p)(t).The first result, whose proof makes use of Chen’s general bijective algorithm for trees and a new decomposition of weakly increasing trees, answers a recent open problem posed by Lin–Ma–Ma–Zhou. The latter result, which we provide both a computational proof and a short combinatorial proof, extends a bi-γ-positivity result due to Ma–Ma–Yeh from p=2 to general p. Their proof for p=2 uses Chen’s context-free grammar and seems hard to be adopted for general p.