论文标题
晶格越过加泰罗尼亚州的系数：$θ_{a} $ - 状态扩展
Coefficients of Catalan States of Lattice Crossing I: $Θ_{A}$-state Expansion
论文作者
论文摘要
J.H.引入了针对平面生根树的多项式挖掘。 przytycki在2014年。正如后来所示，该多项式可用于查找加泰罗尼亚州$ c（a）$ c（a）$ c $ c $ c $ c $ c $ $ m \ times n $ -lattice n $ -lattice crossing $ l（m，n）$，而无需回报。在本文中，我们表明，通过使用$θ_{a} $ - 状态扩展可以找到任何$ c $的$ c（a）$，该扩展代表$ c（a）$作为加泰罗尼亚州系数的线性组合，而加泰罗尼亚州的系数没有$ \ mathbb {q}（q}（a）$。我们还提供了一种用于查找$θ_{a} $的算法 - 状态扩展和其应用程序的示例。最后，给出了具有非抑素系数的加泰罗尼亚州的示例。
Plucking polynomial for plane rooted trees was introduced by J.H. Przytycki in 2014. As it was shown later, this polynomial can be used to find coefficients $C(A)$ of Catalan states $C$ of $m \times n$-lattice crossing $L(m,n)$ without returns on one side. In this paper, we show that $C(A)$ for any $C$ can be found by using $Θ_{A}$-state expansion which represents $C(A)$ as a linear combination of coefficients of Catalan states with no top returns over $\mathbb{Q}(A)$. We also provide an algorithm for finding $Θ_{A}$-state expansions and examples of its applications. Finally, an example of a Catalan state with non-unimodal coefficient is given.
