Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops

TL;DR

This paper introduces determinant formulae for counting tilings of specific domains, linking these counts to the enumeration of Alternating Sign Matrices and Fully Packed Loops, thus advancing combinatorial enumeration techniques.

Contribution

It provides new determinant formulae that connect tiling problems with well-studied combinatorial objects like ASMs and FPLs, offering novel enumeration methods.

Findings

Derived explicit determinant formulae for tiling counts

Established connections between tilings, ASMs, and FPLs

Enhanced enumeration techniques for combinatorial tiling problems

Abstract

We present determinant formulae for the number of tilings of various domains in relation with Alternating Sign Matrix and Fully Packed Loop enumeration.