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{\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.

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