This material has been published in
Adv. Appl. Math.
21 (1998), 381-404,
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Soichi Okada and Christian Krattenthaler
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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