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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