Plucking polynomial of a plane rooted tree with a delay function $\alpha$ was introduced in 2014 by Przytycki. As shown in this paper, plucking polynomial factors when $\alpha$ satisfies additional conditions. We use this result and ${\mathrm{\Theta }}_A$-state expansion introduced in our previous work to derive new properties of coefficients $C(A)$ of Catalan states $C$ resulting from an $(m\times n)$-lattice crossing $L(m,n)$. In particular, we show that $C(A)$ factors when $C$ has arcs with some special properties. In many instances, this yields a more efficient way for computing $C(A)$. As an application, we give closed-form formulas for coefficients of Catalan states of $L(m,3)$.

具有延迟函数的平面根树的采摘多项式$\alpha$由 Przytycki 于 2014 年推出。如本文所示，采摘多项式因子时$\alpha$满足附加条件。我们使用这个结果${\mathrm{\theta }}_A$-我们之前的工作中引入的状态展开来导出系数的新属性$C（A）$加泰罗尼亚州$C$产生于$（米\times n）$-晶格交叉$L（米,n）$。特别是，我们表明$C（A）$当因素$C$具有一些特殊属性的弧。在许多情况下，这产生了一种更有效的计算方式$C（A）$。作为一个应用，我们给出了 Catalan 状态系数的封闭式公式$L（米,3）$。