Enumeration of symmetric centered rhombus tilings of a hexagon

Abstract

A rhombus tiling of a hexagon is said to be centered if it contains the central lozenge. We compute the number of vertically symmetric rhombus tilings of a hexagon with side lengths $a, b, a, a, b, a$ which are centered. When $a$ is odd and $b$ is even, this shows that the probability that a random vertically symmetric rhombus tiling of a $a, b, a, a, b, a$ hexagon is centered is exactly the same as the probability that a random rhombus tiling of a $a, b, a, a, b, a$ hexagon is centered. This also leads to a factorization theorem for the number of all rhombus tilings of a hexagon which are centered.