Exact enumeration of lozenge tilings of a triangular region

2026/07/31 by Jun Yan
Mathematics · #math.CO #msc:05A15 #msc:05B45 #msc:15A15

paper · pdf

12 pages, 5 figures. Corrected error in the statement of Corollary 1.3

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

We prove that the number of lozenge tilings of a certain triangular region T_n is given by the formula T_n=∏_\substack1≤ a<b≤ 3n+2
(a,b)\not=(n+1,2n+2)|1+ζ^a+ζ^b|^1/3, where ζ=e^2πi/(3n+3). This answers a question of Ciucu and Krattenthaler, both by finding the exact formula and by explaining why T_n has many prime factors. The proof reduces the lozenge tiling enumeration problem to evaluating the determinant of the bipartite adjacency matrix M_n of the dual graph of T_n, and then evaluates this determinant by diagonalising M_n.

Citations

• Problems in the Enumeration of Tilings
• Tilings
• Enumeration of lozenge tilings of hexagons with cut off corners
• Trees and Matchings
• Dimer problem in statistical mechanics-an exact result