THE NUMBER OF RHOMBUS TILINGS OF A “PUNCTURED” HEXAGON AND THE MINOR SUMMATION FORMULA

SOICHI OKADA CHRISTIAN KRATTENTHALER

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(⌈\frac{a}{2}⌉, ⌈\frac{b}{2}⌉, ⌈\frac{c}{2}⌉)B(⌈\frac{a+1}{2}⌉, ⌊\frac{b}{2}⌋, ⌈\frac{c}{2}⌉)B(⌈\frac{a}{2}⌉, ⌈\frac{b+1}{2}⌉, ⌊\frac{c}{2}⌋)B(⌊\frac{a}{2}⌋, ⌈\frac{b}{2}⌉, ⌈\frac{c+1}{2}⌉), where B(α, β, γ) is the number of plane partitions inside the α×β×γ 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.