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In Part 1 of the current series of papers, we demonstrate the equivalencebetween the Zhang-Zhang polynomial $\text{ZZ}(\boldsymbol{S},x)$ of aKekul\'ean regular $m$-tier strip $\boldsymbol{S}$ of length $n$ and theextended strict order polynomial $\text{E}_{\mathcal{S}}^{\circ}(n,x+1)$ of acertain partially ordered set (poset) $\mathcal{S}$ associated with$\boldsymbol{S}$. The discovered equivalence is a consequence of the one-to-onecorrespondence between the set $\left\{ K\right\}$ of Kekul\'e structures of$\boldsymbol{S}$ and the set $\left\{\mu:\mathcal{S}\supset\mathcal{A}\rightarrow\left[\,n\,\right]\right\}$ ofstrictly order-preserving maps from the induced subposets of $\mathcal{S}$ tothe interval $\left[\thinspace n\thinspace\right]$. As a result, the problemsof determining the Zhang-Zhang polynomial of $\boldsymbol{S}$ and of generatingthe complete set of Clar covers of $\boldsymbol{S}$ reduce to the problem ofconstructing the set $\mathcal{L}(\mathcal{S})$ of linear extensions of thecorresponding poset $\mathcal{S}$ and studying their basic properties. Inparticular, the Zhang-Zhang polynomial of $\boldsymbol{S}$ can be written in acompact 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 ofdescents and the number of fixed labels, respectively, in the linear extension$w\in\mathcal{L}(\mathcal{S})$.

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