# 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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