# The number of rhombus tilings of a "punctured" hexagon and the minor summation formula

### (21 pages)

Abstract. We compute the number of all rhombus tilings of a hexagon with sides a,b+1,c,a+1,b,c+1, of which the central triangle is removed, provided a,b,c have the same parity. The result is B(\ceil{\frac {a} {2}},\ceil{\frac {b} {2}},\ceil{\frac {c} {2}}) B(\ceil{\frac {a+1} {2}},\floor{\frac {b} {2}},\ceil{\frac {c} {2}}) B(\ceil{\frac {a} {2}},\ceil{\frac {b+1} {2}},\floor{\frac {c} {2}}) B(\floor{\frac {a} {2}},\ceil{\frac {b} {2}},\ceil{\frac {c+1} {2}}), where B(a,b,c) is the number of plane partitions inside the a x b x c box. The proof uses nonintersecting lattice paths and a new identity for Schur functions, which is proved by means of the minor summation formula of Ishikawa and Wakayama. A symmetric generalization of this identity is stated as a conjecture.

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