##
Mihai Ciucu
and Christian Krattenthaler

# Plane partitions II: 5 1/2 symmetry classes

### (18 pages)

**Abstract.**
We present new, simple proofs for the enumeration of five of the ten
symmetry classes of plane partitions contained in a given box. Four of
them are derived from a simple determinant evaluation, using
combinatorial arguments. The previous proofs of these four cases were
quite complicated. For one
more symmetry class we give an elementary proof in the case when two of the
sides of the box are equal. Our results include simple evaluations of
the determinants det ( *\delta*_{ij} + *\binom*
{*x+i+j*} {*i*} )
_{0<=i,j<=n-1} and
det (* \binom *{*x+i+j*} {*2j-i*} )
_{0<=i,j<=n-1}, notorious in plane partitions enumeration, whose
previous evaluations were quite intricate.

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