{\it tiered graph} $G=(V,E)$ in $m $ stage is a simple graph of $V\subseteq \brk{n}$. where $ If v$ is a vertex adjacent to $v’$ in $G$, then $t(v) >t(v’)$ if v>v’$. For any ordered partition $p=(p_1,p_2,\cdots,p_m)$ of $n$, $\sett_p$ is stratified with vertex set $\brk{n}$ and map $t. Suppose we represent a set of trees with : \brk{n}\rightarrow \brk{m}$ such that $|t^{-1}(i)|=p_i$ for all $i=1,2,\ldots,m$. For any $T\in \sett_p$, let $K_T$ represent the complete layered graph whose vertex sets and layered maps are the same as those of $T$. If the edges of $K_T$ are lexicographically ordered 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 $e\in is. Let E(K_{T})\setminus E(T)$ let $e$ be the smallest element of the unique cycle determined by $T\cup e$. Let $P_p(q)=\sum_{T\in \sett_{p}}q^{w(T)}$. Dugan, Glennon, Gunnels, Steingr’imsson [J. Combin. Theory, Ser. A 164 (2019) pp. 24-49] $\pi(p )=p_{\pi(1)},p_{\pi(2)},\cdots,p_{\pi(m)})$. We will prove an extension of this identity. We also provide a proof of the identity $P_{(1,p_1,p_2)}(q)=P_{(p_1+1,p_2+1)}(q)$ via the Tutte polynomials.

Source link


Leave A Reply