MathDB

Di Francesco's determinant evaluation for twenty-vertex configurations

About 2 years old · traced to

Let nn be a positive integer. For integers α\alpha and pp, define

(αp)={α(α−1)⋯(α−p+1)p!,p≥0,0,p<0.\binom{\alpha}{p}=\begin{cases}\dfrac{\alpha(\alpha-1)\cdots(\alpha-p+1)}{p!},&p\geq0,\\0,&p<0.\end{cases}

Di Francesco's conjecture.

det⁡0≤i,j≤n−1(2i(i+2j+12j+1)−(i−12j+1))=2∏i=1n2i−1(4i−2)!(n+2i−1)!.\det_{0\leq i,j\leq n-1}\left(2^i\binom{i+2j+1}{2j+1}-\binom{i-1}{2j+1}\right)=2\prod_{i=1}^n\frac{2^{i-1}(4i-2)!}{(n+2i-1)!}.

This determinant counts the relevant twenty-vertex configurations and corresponding domino tilings of Aztec triangles. The source paper proves the conjecture, although the supplied parser does not provide a resolution status; the claim is therefore recorded as open pending verification of that proof.

References

Primary source

Christoph Koutschan, Christian Krattenthaler and Michael Schlosser, “Determinant evaluations inspired by Di Francesco's determinant for twenty-vertex configurations”, arXiv:2401.08481 (2024).

determinantsdomino-tilingstwenty-vertex-model
Posted

Progress summary

Refreshed
Claimed solved

A 2024 paper claims to prove the formula, and a 2025 paper gives a separate proof of the equivalent tiling count; this scan found no challenge, but verification is pending.

Di Francesco’s conjecture evaluates a determinant counting twenty-vertex configurations and domino tilings of Aztec triangles. The result was announced by Doron Zeilberger on June 25, 2021, after Christoph Koutschan obtained a proof.

Reported proofs, 2024–2025

Koutschan, Krattenthaler, and Schlosser’s 2024 paper proves the stated identity using Zeilberger’s holonomic Ansatz, with computational details in accompanying material. A 2025 paper gives a separate combinatorial proof of the corresponding Aztec-triangle tiling formula. The retrieved sources report no counterexample, withdrawal, or proof-gap objection; this card nevertheless treats the resolution as unverified.

Current status (as of September 2026): The determinant identity is claimed proved by Koutschan, Krattenthaler, and Schlosser, with an independent related tiling proof, but formal verification of the reported resolution remains pending.

Sources

Solutions 0

No solutions have been posted yet.