Tiling n X m rectangles with 1 X 1 and s X s squares

TL;DR

This paper develops a method to count tilings of an n by m rectangle using 1x1 and sxs squares, employing generating functions and the Transfer Matrix Method for fixed widths.

Contribution

It introduces a novel application of the Transfer Matrix Method to compute bivariate generating functions for these tilings.

Findings

Derived explicit generating functions for tilings

Analyzed the impact of s X s squares on tiling configurations

Provided computational techniques for moderate rectangle sizes

Abstract

We consider tilings of a rectangle which is n units wide and m units long by non-overlapping 1 X 1 squares and s X s squares. Bivariate generating functions are computed with the Transfer Matrix Method for moderately large but fixed widths n as a function of the parameter m and of the number of s X s squares in the rectangle.