# (-1)-enumeration of plane partitions with complementation symmetry

## Theresia Eisenkölbl

### Abstract

We compute the weighted enumeration of plane
partitions contained in a given box with complementation symmetry
where adding one half of an orbit of cubes and removing the other half
of the orbit changes the weight by
-1 as proposed by Kuperberg. We use
nonintersecting lattice path
families to accomplish this for transpose-complementary,
cyclically symmetric transpose-complementary and totally symmetric
self-complementary plane
partitions. For symmetric
transpose-complementary and self-complementary
plane partitions we get partial results. We also describe Kuperberg's
proof for the case of
cyclically symmetric self-complementary plane partitions.
(41 pages)

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