ZZ Polynomials of Regular m-tier Benzenoid Strips as Extended Strict Order Polynomials of Associated Posets Part 2. Guide to Practical Computation

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 ZZ(S, x) of a regular benzenoid strip S and the extended strict order polynomial E-S degrees(n, 1 + x) of the corresponding poset S, demonstrated formally in Part 1 of the current series of papers. The process of computing ZZ(S, x) in the form of E-S degrees(n, 1 + x) reduces to four, fully automatable steps: (i) Construction of the poset S corresponding to S. (ii) Construction of the Jordan-Holder set L(S) of linear extensions of S. (iii) Computing the number des(w) of descents in each w is an element of L(S). (iv) Computing the number fix(S)(w) of fixed labels in each w is an element of L(S). The ZZ polynomial of S can then be expressed in the following form ZZ(S, x) = E-S degrees(n, 1 + x) = Sigma(w is an element of L(S))Sigma(vertical bar S vertical bar)(k=0)(vertical bar S vertical bar k fix(S)(w) - fix(S)(w)) (n + des(w) k)(1 + x)(k), where vertical bar S vertical bar denotes the number of elements in S. Practical applications of the algorithm are illustrated with a few examples. The complete account of ZZ polynomials of regular m-tier benzenoid strips S with m = 1- 6 computed using the introduced algorithm is presented in Part 3 of the current series of papers.

Automated summary

The algorithm presented in this paper calculates the ZZ polynomial of regular strips through four steps, including constructing the corresponding poset, constructing the set of linear extensions, computing the number of descents, and computing the number of fixed labels. The practical applications of the algorithm are demonstrated with several examples, showcasing the computation of ZZ polynomials for regular m-tier benzenoid strips.