Enumeration of symmetric centered rhombus tilings of a hexagon
(48 pages)
Abstract.
A rhombus tiling of a hexagon is said to be centered
if it contains the central rhombus. 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.
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