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

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 ZZ(S, x) of a regular benzenoid strip S and the extended strict order polynomial E • S (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 (n, 1 + x) reduces to four, fully automatable steps: (i) Construction of the poset S corresponding to S. (ii) Construction of the Jordan-Hölder set L(S) of linear extensions of S. (iii) Computing the number des(w) of descents in each w ∈ L(S). (iv) Computing the number fix S (w) of fixed labels in each w ∈ L(S). The ZZ polynomial of S can then be expressed in the following formwhere |S| 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.

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

“…the process of determination of ZZ(S, x) can be completely automatized for any regular m-tier strip S. The corresponding algorithm, whose details are elaborated in Part 2 of the current series of papers [43] (see also [40] for mathematical details), can be summarized by and an arbitrary value of n using the corresponding posets S and Eq. ( 14) is presented in Part 3 of the current series of papers [44].…”

“…Practical application of Theorem 24 to the determination of ZZ polynomials of regular strips is presented in Parts 2 and 3 in this series of papers [43,44], where we give a practical guide to computation of the extended strict order polynomials E • S (n, 1 + x) together with a complete account of ZZ polynomials ZZ(S, x) of regular m-tier benzenoid strips S with m = 1-6 and an arbitrary value of n determined as the extended strict order polynomials E • S (n, 1 + x) of the corresponding posets S. It would be inconvenient to present this collection of results here owing to its somewhat bulky volume. However, in order to foreshadow the forthcoming results, we illustrate very briefly the process of determination of E • S (n, 1+x) using Eq.…”