格点交叉II的Catalan态系数：$Θ_{A}$ -态展开的应用
Coefficients of Catalan States of Lattice Crossing II: Applications of $Θ_{A}$-state Expansions
摘要：
具有延迟函数$α$的平面有根树的采摘多项式由J.H.~ Przytycki。本文证明了当$α$满足附加条件时，多项式的拔出因子.我们使用这个结果和在我们以前的工作中介绍的$Θ_{A}$ -状态展开导出新的性质的系数$C(A)$的加泰罗尼亚状态$C$产生的$m \times n$ -晶格交叉$L(m,n)$。特别是，我们表明，$C(A)$因素时，$C$有一些特殊性质的弧。在许多情况下，这产生用于计算$C(A)$的更有效的方式。作为应用，我们给出了$L(m,3)$的Catalan态系数的封闭式公式。
英文摘要：
Plucking polynomial of a plane rooted tree with a delay function $α$ was introduced in 2014 by J.H.~Przytycki. As shown in this paper, plucking polynomial factors when $α$ satisfies additional conditions. We use this result and $Θ_{A}$-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 $L(m,3)$.
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