Automatic Counting of Tilings of Skinny Plane Regions

TL;DR

This paper introduces an inductive approach to derive explicit generating functions for counting dimer and monomer-dimer tilings in large, narrow plane regions, marking a shift from traditional deductive methods.

Contribution

It is the first to use inductive methods to rigorously find generating functions for tilings of specific narrow plane regions.

Findings

Derived explicit generating functions for dimer tilings.

Extended methods to monomer-dimer tilings.

Validated the approach on large families of regions.

Abstract

The deductive method ruled mathematics for the last 2500 years, now it is the turn of the inductive method. Here we make a modest start by using the inductive method to discover and prove (rigorously) explicit generating functions for the number of dimer (and monomer-dimer) tilings of large families of "skinny" plane regions.