Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2508.04545

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

Seok Hyun Byun·Mihai Ciucu
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.COcond-mat.stat-mechmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Seok Hyun Byun, Mihai Ciucu. 2025-08-06. A short combinatorial proof of Di Francesco's conjecture on Aztec triangles. https://arxiv.org/abs/2508.04545

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗
arXiv2026

Generalized Goulden-Yong duals and signed minimal factorizations

In this paper, we give two combinatorial ways to study signed exceptional sequences. First, we show the equivalence between one-way reflections and relatively projective representations. Secondly, we construct generalized Goulden-Yong duals using reverse Garside element actions and folded chord diagrams. We then give two applications of the generalized Goulden-Yong duals: constructing generalized Prüfer codes and counting signed factorizations using the matrix-tree theorem.

math.CO↗
arXiv2026

Explicit expressions for iterates of power series

We present several formulas for both the discrete and fractional iterates of an invertible power series $f$, using a new unifying approach based on umbral calculus. Known formulas are extended, and their proofs simplified, while new expressions are introduced. In particular, by employing $q$-calculus identities, we eliminate the requirement for $f'(0)$ to equal $1$ and the resulting general expressions for the iterative logarithm are obtained as well.

math.CO↗