  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.
