The number of rhombus tilings of a "punctured" hexagon and the minor summation formula
Soichi Okada and Christian Krattenthaler
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 $B(\lceil{\frac {a} {2}}\rceil,\lceil{\frac {b} {2}}\rceil,\lceil{\frac {c} {2}}\rceil)$ $B(\lceil{\frac {a+1} {2}}\rceil,\lfloor{\frac {b} {2}}\rfloor,\lceil{\frac {c} {2}}\rceil)$ $B(\lceil{\frac {a} {2}}\rceil,\lceil{\frac {b+1} {2}}\rceil,\lfloor{\frac {c} {2}}\rfloor)$ $B(\lfloor{\frac {a} {2}}\rfloor,\lceil{\frac {b} {2}}\rceil,\lceil{\frac {c+1} {2}}\rceil)$, where $B(\alpha,\beta,\gamma)$ is the number of plane partitions inside the $\alpha\times \beta\times \gamma$ 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.