##
Christian Krattenthaler

# A (conjectural) 1/3-phenomenon for the number of rhombus
tilings of a hexagon which contain a fixed rhombus

### (16 pages)

**Abstract.**
We state, discuss, provide evidence for, and prove in special cases
the conjecture that the
probability that a random tiling by rhombi of a
hexagon with side lengths
2*n*+*a*,2*n*+*b*,2*n*+*c*,2*n*+*a*,2*n*+*b*,2*n*+*c*
contains
the (horizontal) rhombus with coordinates
(2*n*+*x*,2*n*+*y*) is equal to
$*1/3 +
g_{a,b,c,x,y}(n){\binom {2n}{n}}^3/\binom {6n}{3n}*$,
where *g*_{a,b,c,x,y}(*n*) is a rational function in *n*.
Several specific
instances of this "1/3-phenomenon" are made explicit.

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