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
2n+a,2n+b,2n+c,2n+a,2n+b,2n+c
contains
the (horizontal) rhombus with coordinates
(2n+x,2n+y) is equal to
$1/3 +
g_{a,b,c,x,y}(n){\binom {2n}{n}}^3/\binom {6n}{3n}$,
where ga,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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