Proving identities on weight polynomials of tiered trees via Tutte polynomials

Abstract

A {\it tiered graph} $G=(V,E)$ with $m $ tiers is a simple graph with $V\subseteq \brk{n}$, where $\brk{n}=\{1,2,\cdots,n\}$, and with a surjective map $t$ from $V$ to $\brk{m}$ such that if $v$ is a vertex adjacent to $v'$ in $G$ with $v>v'$, then $t(v) >t(v')$. For any ordered partition $p=(p_1,p_2,\cdots,p_m)$ of $n$, let $\sett_p$ denote the set of tiered trees with vertex set $\brk{n}$ and with a map $t: \brk{n}\rightarrow \brk{m}$ such that $|t^{-1}(i)|=p_i$ for all $i=1,2,\ldots,m$. For any $T\in \sett_p$, let $K_T$ denote the complete tiered graph whose vertex set and tiering map are the same as those of $T$. If the edges of $K_T$ are ordered lexicographically by their endpoints, then the weight $w(T)$ of $T$ is the external activity of $T$ in $K_T$, i.e., the number of edges $e\in E(K_{T})\setminus E(T)$ such that $e$ is the least element in the unique cycle determined by $T\cup e$. Let $P_p(q)=\sum_{T\in \sett_{p}}q^{w(T)}$. Dugan, Glennon, Gunnells and Steingr\'imsson [J. Combin. Theory, Ser. A 164 (2019) pp. 24-49] asked for an elementary proof of the identity $P_p(q)=P_{\pi(p)}(q)$ for any permutation $\pi$ of $1,2,\cdots,m$, where $\pi(p)=p_{\pi(1)},p_{\pi(2)},\cdots,p_{\pi(m)})$. In this article, we will prove an extension of this identity by applying Tutte polynomials. Furthermore, we also provide a proof of the identity $P_{(1,p_1,p_2)}(q)=P_{(p_1+1,p_2+1)}(q)$ via Tutte polynomials.