Proving identities on weight polynomials of tiered trees via Tutte polynomials

Abstract

A tiered graphG=(V,E) with m tiers is a simple graph with V⊆〚n〛, where 〚n〛={1,2,⋯,n}, and with a surjective map t from V to 〚m〛 such that if v is a vertex adjacent to v′ in G with v>v′, then t(v)>t(v′). For any ordered partition p=(p1,p2,⋯,pm) of n, let Tp denote the set of tiered trees with vertex set 〚n〛 and with a map t:〚n〛→〚m〛 such that |t−1(i)|=pi for all i=1,2,…,m. For any T∈Tp, let KT denote the complete tiered graph whose vertex set and tiering map are the same as those of T. If the edges of KT are ordered lexicographically by their endpoints, then the weight w(T) of T is the external activity of T in KT, i.e., the number of edges e∈E(KT)∖E(T) such that e is the least element in the unique cycle determined by T∪e. Let Pp(q)=∑T∈Tpqw(T). Dugan, Glennon, Gunnells and Steingrímsson (2019) [4] asked for an elementary proof of the identity Pp(q)=Pπ(p)(q) for any permutation π of 1,2,⋯,m, where π(p)=(pπ(1),pπ(2),⋯,pπ(m)). In this article, we will prove an extension of this identity by applying Tutte polynomials. Furthermore, we also provide a proof of the identity P(1,p1,p2)(q)=P(p1+1,p2+1)(q) via Tutte polynomials.