TY - JOUR

T1 - ZZ Polynomials of Regular m-tier Benzenoid Strips as Extended Strict Order Polynomials of Associated Posets Part 2. Guide to Practical Computation

AU - Langner, Johanna

AU - Witek, Henryk A.

N1 - Publisher Copyright:
© 2022 University of Kragujevac, Faculty of Science. All rights reserved.

PY - 2022

Y1 - 2022

N2 - 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 fixS(w) of fixed labels in each w ∈ L(S). The ZZ polynomial of S can then be expressed in the following form ∑ w∈L(S)∑k|S|=0(|S| k − − fix fix S S ((w w ) ))(n + des(k w)) (1 + x)k, ZZ(S, x) = E◦S(n, 1 + x) = where |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.

AB - 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 fixS(w) of fixed labels in each w ∈ L(S). The ZZ polynomial of S can then be expressed in the following form ∑ w∈L(S)∑k|S|=0(|S| k − − fix fix S S ((w w ) ))(n + des(k w)) (1 + x)k, ZZ(S, x) = E◦S(n, 1 + x) = where |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.

UR - http://www.scopus.com/inward/record.url?scp=85127796833&partnerID=8YFLogxK

U2 - 10.46793/match.88-1.109L

DO - 10.46793/match.88-1.109L

M3 - Article

AN - SCOPUS:85127796833

SN - 0340-6253

VL - 88

SP - 109

EP - 130

JO - Match

JF - Match

IS - 1

ER -