  Plucking polynomial of a plane rooted tree with a delay function $\alpha$ was
introduced in 2014 by J.H.~Przytycki. As shown in this paper, plucking
polynomial factors when $\alpha$ satisfies additional conditions. We use this
result and $\Theta_{A}$-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
$L(m,3)$.
