The Number of Centered Lozenge Tilings of a Symmetric Hexagon

摘要

Propp conjectured that the number of lozenge tilings of a semiregular hexagon of sides 2n - 1, 2n - 1, and 2n which 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 sides a, a, and b. We prove explicit formulas for the number of lozenge tilings of these hexagons containing the central unit rhombus and obtain Propp's conjecture as a corollary of our results.