Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards

TL;DR

This paper develops a mathematical framework using generating functions and transfer matrix methods to count non-overlapping non-bonding domino configurations on rectangular boards, providing exact solutions for small board sizes.

Contribution

It introduces a novel enumeration approach for non-bonding dominoes using bivariate generating functions and transfer matrices, extending combinatorial enumeration techniques.

Findings

Derived rational generating functions for small boards

Enumerated configurations for boards with up to six rows or columns

Provided exact formulas for non-bonding domino arrangements

Abstract

The manuscript studies configurations of non-overlapping non-bonding dominoes on finite rectangular boards of unit squares characterized by row and column number. The non-bonding dominoes are defined here by the requirement that any domino on the board shares at most one point (one of its four corner points) with any other domino, but no edge. With the Transfer Matrix Method, rational generating functions are derived that solve the enumeration problem entirely, here evaluated for boards with up to six rows or columns.