Cruciform regions and a conjecture of Di Francesco

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.