Weakly increasing trees on a multiset

Abstract

In this paper, we introduce the concept of weakly increasing trees on a multiset M, which is an extension of plane trees and increasing trees on the set {0,1,…,n}. We define the M-Eulerian–Narayana polynomial for weakly increasing trees on a multiset M, which interpolates between the Eulerian polynomial and the Narayana polynomial. We obtain a compact product formula for the number of weakly increasing trees on a general multiset. Inspired by some remarkable equidistributions between multipermutations and s-inversion sequences, we establish two connections between our M-Eulerian–Narayana polynomials for the multiset M={12,22,…,n2} (resp. M={12,22,…,(n−1)2,n}) and Savage and Schuster's s-Eulerian polynomials for the sequence s=(1,1,3,2,5,3,…,2n−1,n,n+1) (resp. s=(1,1,3,2,5,3,…,2n−1,n)). We also derive equidistributions that involve some natural tree statistics among weakly increasing trees on different multisets. Via introducing a group action on weakly increasing trees, we prove combinatorially the γ-positivity of the M-Eulerian–Narayana polynomials for a general multiset M. As an application of this γ-positivity result, we obtain a combinatorial interpretation of γ-coefficients of descent polynomials of permutations on the multiset {12,22,…,n2} in terms of weakly increasing trees.