Problems in the Enumeration of Tilings

TL;DR

This paper surveys the field of tiling enumeration, discusses progress on longstanding open problems, and introduces new tiling problems to advance research in algebraic combinatorics.

Contribution

It reviews two decades of progress on open tiling problems and proposes new challenges to stimulate further research in the field.

Findings

Most of Propp's 1999 problems have been solved or generalized

The paper introduces new tiling problems for future exploration

Highlights the vibrant connections of tiling enumeration to other mathematical areas

Abstract

Enumeration of tilings is the mathematical study concerning the total number of coverings of regions by similar pieces without gaps or overlaps. Enumeration of tilings has become a vibrant subfield of combinatorics with connections and applications to diverse mathematical areas. In 1999, James Propp published his well-known list of 32 open problems in the field. The list has got much attention from experts around the world. After two decades, most of the problems on the list have been solved and generalized. In this paper, we propose a set of new tiling problems. This survey paper contributes to the Open Problems in Algebraic Combinatorics 2022 conference (OPAC 2022) at the University of Minnesota.