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

Abstract

In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial ZZ(S, x) of a Kekuléan regular m-tier strip S of length n and the extended strict order polynomial E • S (n, x + 1) of a certain partially ordered set (poset) S associated with S. The discovered equivalence is a consequence of the one-to-one correspondence between the set {K} of Kekulé structures of S and the set {µ : S ⊃ A → [ n ]} of strictly order-preserving maps from the induced subposets of S to the interval [ n ]. As a result, the problems of determining the Zhang-Zhang polynomial of S and of generating the complete set of Clar covers of S reduce to the problem of constructing the set L(S) of linear extensions of the corresponding poset S and studying their basic properties. In particular, the Zhang-Zhang polynomial of S can be written in a compact form as ZZ(S, x) = |S| k=0 w∈L(S)