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

Abstract

In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial $\text{ZZ}(\mathbf{S},x)$ of a Kekuléan regular $m$-tier strip $\mathbf{S}$ 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 $\mathbf{S}$. The discovered equivalence is a consequence of the one-to-one correspondence between the set $\left\{K\right\}$ of Kekulé structures of $\mathbf{S}$ and the set $\left\{\mu :\mathcal{S}\supset \mathcal{A}\to \left[n\right]\right\}$ of strictly order-preserving maps from the induced subposets of $\mathcal{S}$ to the interval $\left[n\right]$. As a result, the problems of determining the Zhang-Zhang polynomial of $\mathbf{S}$ and of generating the complete set of Clar covers of $\mathbf{S}$ 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 $\mathbf{S}$ can be written in a compact form as $\text{ZZ}(\mathbf{S},x)={\sum }_{k=0}^{|\mathcal{S}|}{\sum }_{w\in \mathcal{L}(\mathcal{S})}\left(\genfrac{}{}{0ex}{}{|\mathcal{S}|-{\text{fix}}_{\mathcal{S}}(w)}{k-{\text{fix}}_{\mathcal{S}}(w)}\right)\left(\genfrac{}{}{0ex}{}{n+\text{des}(w)}{k}\right){\left(1+x\right)}^k$, 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 $w\in \mathcal{L}(\mathcal{S})$.

36 pages, submitted to MATCH Commun. Math. Comput. Chem

References in corpus (1)

• Equivalence of the Generalized Zhang-Zhang Polynomial and the Generalized Cube Polynomial