ZZ Polynomials of Regular $m$-tier Benzenoid Strips as Extended Strict Order Polynomials of Associated Posets -- Part 1. Proof of Equivalence

TL;DR

This paper establishes a mathematical equivalence between the Zhang-Zhang polynomial of regular m-tier benzenoid strips and extended strict order polynomials of associated posets, simplifying the calculation of Kekulé structures.

Contribution

It proves the equivalence between the Zhang-Zhang polynomial and extended strict order polynomials, linking Kekulé structures to linear extensions of posets.

Findings

Zhang-Zhang polynomial can be expressed via linear extensions of associated posets.

The equivalence simplifies the enumeration of Kekulé structures and Clar covers.

Provides a compact formula for the Zhang-Zhang polynomial in terms of poset properties.

Abstract

In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial $\text{ZZ}(\boldsymbol{S},x)$ of a Kekul\'ean regular $m$-tier strip $\boldsymbol{S}$ of length $n$ and the extended strict order polynomial $\text{E}_{\mathcal{S}}^{\circ}(n,x+1)$ of a certain partially ordered set (poset) $\mathcal{S}$ associated with $\boldsymbol{S}$. The discovered equivalence is a consequence of the one-to-one correspondence between the set $\left\{ K\right\}$ of Kekul\'e structures of $\boldsymbol{S}$ and the set $\left\{ \mu:\mathcal{S}\supset\mathcal{A}\rightarrow\left[\,n\,\right]\right\}$ of strictly order-preserving maps from the induced subposets of $\mathcal{S}$ to the interval $\left[\thinspace n\thinspace\right]$. As a result, the problems of determining the Zhang-Zhang polynomial of $\boldsymbol{S}$ and of generating the complete set of Clar covers of $\boldsymbol{S}$ reduce to the problem of constructing the set $\mathcal{L}(\mathcal{S})$ of linear extensions of the corresponding poset $\mathcal{S}$ and studying their basic properties. In particular, the Zhang-Zhang polynomial of $\boldsymbol{S}$ can be written in a compact form as $\text{ZZ}(\boldsymbol{S},x)=\sum_{k=0}^{\left|\mathcal{S}\right|}\sum_{w\in\mathcal{L}(\mathcal{S})}\binom{\left|\mathcal{S}\right|-\text{fix}_{\mathcal{S}}(w)}{\,\,k\,\,\hspace{1pt}-\text{fix}_{\mathcal{S}}(w)}\binom{n+\text{des}(w)}{k}\left(1+x\right)^{k}$, where $\text{des}(w)$ and $\text{fix}_{\mathcal{S}}(w)$ denote the number of descents and the number of fixed labels, respectively, in the linear extension $w\in\mathcal{L}(\mathcal{S})$.