Cruciform regions and a conjecture of Di Francesco

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 Tn\mathcal {T}_n, are obtained by starting with a square of side-length 2n2n, 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−1n-1. Inspired by the regions Tn\mathcal {T}_n, we construct a family Cm,na,b,c,dC_{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 Tn\mathcal {T}_n is a divisor of the number of tilings of the cruciform region C2n−1,2n−1n−1,n,n,n−2C_{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.