Coefficients of Catalan states of lattice crossing II: Applications of ΘA-state expansions

Abstract

Plucking polynomial of a plane rooted tree with a delay function [Formula: see text] was introduced in 2014 by Przytycki. As shown in this paper, plucking polynomial factors when [Formula: see text] satisfies additional conditions. We use this result and [Formula: see text]-state expansion introduced in our previous work to derive new properties of coefficients [Formula: see text] of Catalan states [Formula: see text] resulting from an [Formula: see text]-lattice crossing [Formula: see text]. In particular, we show that [Formula: see text] factors when [Formula: see text] has arcs with some special properties. In many instances, this yields a more efficient way for computing [Formula: see text]. As an application, we give closed-form formulas for coefficients of Catalan states of [Formula: see text].