A short combinatorial proof of Di Francesco's conjecture on Aztec triangles

2025/08/06 by Byun, Seok Hyun, Ciucu, Mihai
#05A15 #82B20 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2508.04545

Abstract

Di Francesco conjectured in 2021 that the number of domino tilings of a certain family of regions -- called Aztec triangles -- on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. This turned out to be a real challenge to prove without the use of computers -- each of the two existing proofs (one due to Koutschan, Krattenthaler and Schlosser, the other to Corteel, Huang and Krattenthaler) relies on substantial computer calculations which would be hard to check directly. In this paper we present a short combinatorial proof that relies on the second author's factorization theorem and complementation theorem for perfect matchings.

Citations

• Domino tilings of generalized Aztec triangles
• Cruciform regions and a conjecture of Di Francesco
• Twenty Vertex model and domino tilings of the Aztec triangle
• Enumeration of tilings of diamonds and hexagons with defects
• Another proof of the alternating sign matrix conjecture
• Schur function identities and the number of perfect matchings of holey Aztec rectangles

Related

• Domino tilings of generalized Aztec triangles
• Cruciform regions and a conjecture of Di Francesco
• A generalization of Aztec diamond theorem, part I
• Twenty Vertex model and domino tilings of the Aztec triangle
• Symmetric Domino Tilings of Aztec Diamonds