Abstract
We present an algorithm for computing the ZZ polynomial of an arbitrary \(m\)-tier regular strip of length \(n\). Our approach is based on the equivalence between the ZZ polynomial \(\text{ZZ}(\boldsymbol{S},x)\) of a regular benzenoid strip \(\boldsymbol{S}\)
and the extended strict order polynomial \(\text{E}_{\mathcal{S}}^{\circ}(n,1+x)\) of the corresponding poset \(\mathcal{S}\), demonstrated formally in Part 1 of the current series of papers. The process of computing \(\text{ZZ}(\boldsymbol{S},x)\)
in the form of \(\text{E}_{\mathcal{S}}^{\circ}(n,1+x)\) reduces to four, fully automatable steps: \(\left(i\right)\)~Construction of the poset \(\mathcal{S}\) corresponding to \(\boldsymbol{S}\). \(\left(ii\right)\)~Construction of the
Jordan-Hölder set \(\mathcal{L}(\mathcal{S})\) of linear extensions of \(\mathcal{S}\). \(\left(iii\right)\)~Computing the number \(\operatorname{des}(w)\) of descents in each \(w\in\mathcal{L}(\mathcal{S})\). \(\left(iv\right)\)~Computing
the number \(\operatorname{fix}_{\mathcal{S}}(w)\) of fixed labels in each \(w\in\mathcal{L}(\mathcal{S})\). The ZZ polynomial of \(\boldsymbol{S}\) can then be expressed in the following form \[ \text{ZZ}(\boldsymbol{S},x)=\text{E}_{\mathcal{S}}^{\circ}(n,1+x)=\sum_{w\in\mathcal{L}(\mathcal{S})}\sum_{k=0}^{\left|\mathcal{S}\right|}\binom{\left|\mathcal{S}\right|-\operatorname{fix}_{\mathcal{S}}(w)}{\,\,k\,\,\hspace{1pt}-\operatorname{fix}_{\mathcal{S}}(w)}\binom{n+\operatorname{des}(w)}{k}\left(1+x\right)^{k},
\] where \(\left|\mathcal{S}\right|\) denotes the number of elements in \(\mathcal{S}\). Practical applications of the algorithm are illustrated with a few examples. The complete account of ZZ polynomials of regular \(m\)-tier benzenoid
strips \(\boldsymbol{S}\) with \(m=1-6\) computed using the introduced algorithm is presented in Part 3 of the current series of papers.
