  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
$w\in\mathcal{L}(\mathcal{S})$.
