Research ArticleNo Access

Coefficients of Catalan states of lattice crossing II: Applications of $Θ_{A}$ -state expansions

Mieczyslaw K. Dabkowski

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA

E-mail Address: mdab@utdallas.edu

Corresponding author.

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and

Cheyu Wu

https://orcid.org/0000-0002-1490-9843

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA

E-mail Address: cheyu.wu@utdallas.edu

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https://doi.org/10.1142/S0218216524500032Cited by:0 (Source: Crossref)

Abstract

Plucking polynomial of a plane rooted tree with a delay function $α$ was introduced in 2014 by 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 $L (m, 3)$ .

Keywords:

AMSC: 57K10, 57K31

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