Cruciform regions and a conjecture of Di Francesco

2021/04/19 by Ciucu, Mihai · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2104.09405

Abstract

A recent conjecture of Di Francesco states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted \mathcal T_n, are obtained by starting with a square of side-length 2n, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order n-1. Inspired by the regions \mathcal T_n, we construct a family C_m,n^a,b,c,d of cruciform regions generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of \mathcal T_n is a divisor of the number of tilings of the cruciform region C_2n-1,2n-1^n-1,n,n,n-2, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture.

Cited by

• Round Aztec windows, a dual of the Aztec diamond theorem and a curious symmetry of the correlation of diagonal slits
• A short combinatorial proof of Di Francesco's conjecture on Aztec triangles

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• A simple proof for the number of tilings of quartered Aztec diamonds