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

TL;DR

This paper provides a concise combinatorial proof confirming Di Francesco's 2021 conjecture that the number of domino tilings of Aztec triangles follows a specific product formula, avoiding extensive computer calculations.

Contribution

It introduces a new short combinatorial proof using factorization and complementation theorems, simplifying the verification of Di Francesco's conjecture.

Findings

Confirmed the conjecture with a combinatorial proof

Simplified the proof process compared to previous computer-based methods

Validated the product formula for Aztec triangle tilings

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.