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 T n \mathcal {T}_n , are obtained by starting with a square of side-length 2 n 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 n-1 . Inspired by the regions T n \mathcal {T}_n , we construct a family C m , n a , b , c , d 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 T n \mathcal {T}_n is a divisor of the number of tilings of the cruciform region C 2 n − 1 , 2 n − 1 n − 1 , n , n , n − 2 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.
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