##### 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.