The Number of Centered Lozenge Tilings of a Symmetric Hexagon

Propp conjectured that the number of lozenge tilings of a semiregular hexagon of sides 2n−1, 2n−1, and 2nwhich contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregular hexagon of sidesa,a, andb. We prove explicit formulas for the number of lozenge tilings of these hexagons containing the central unit rhombus and obtain Proppʹs conjecture as a corollary of our results.