I'm reading an article named Selfinjective and simply connected algebras written by Otto Bretscher, Christian L$\ddot{\text{a}}$ser and Christine Riedtmann in 1981. Here is the article: https://link.springer.com/article/10.1007/BF01322494
In 1.1, the authors wrote that for every point $x$ in $\mathbb{Z}\Delta$, there is a non-zero path in mesh category $k(\mathbb{Z}\Delta):$ $$v^{-1}(x)\rightarrow \dots \rightarrow x,$$ where $v$ is the Nakayama-permutation of $\mathbb{Z}\Delta$. Moreover, this path is the longest among all the paths with end $x$, and the length of the longest path is denoted as $m_{\Delta}$. The authors claimed that $$m_{A_n}=n,m_{D_n}=2n-3,m_{E_6}=11,m_{E_7}=17,m_{E_8}=29$$ But I do not know how to prove the path $v^{-1}(x)\rightarrow \dots \rightarrow x$ is the longest non-zero path in $k(\mathbb{Z}\Delta)$.
The definition of Nakayama-permutation $v:$
1.$\Delta=A_n:v(p,q)=(p+q+1,n+1-q);$
2.$\Delta=D_{2k}:v(p,q)=(p+2k-2,q)$;
3.$\Delta=D_{2k+1}:$\begin{equation*}v(p,q)=\begin{cases}(p+2k-1,q),&q\leqslant 2k-1;\\ (p+2k-1,2k+1),&q=2k;\\ (p+2k-1,2k),&q=2k+1;\end{cases}\end{equation*}
4.$\Delta=E_6:$ \begin{equation*}v(p,q)=\begin{cases}(p+5,6-q),&q\leqslant 5 ;\\(p+5,6),&q=6;\end{cases}\end{equation*}
5.$\Delta=E_7:v(p,q)=(p+8,q)$;
6.$\Delta=E_8:v(p,q)=(p+14,q)$.