This material has been published in Adv. Appl. Math. 21 (1998), 381-404, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Elsevier B.V. This material may not be copied or reposted without explicit permission.

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.

The following versions are available:

Back to Christian Krattenthaler's home page.