Macdonald's plane partition conjecture proposes a formula for the number of cyclically symmetric plane partitions (CSPPs)
of a given integer whose Ferrers
diagrams fit inside an 
box. Macdonald (1982) gave a product
representation for the power series whose coefficients
of 
were the number of such partitions
of 
.
Let 
be the set of all integer points 
in the first octant such
that a plane partition 
is defined and 
. Then 
is said to be cyclically
symmetric if 
is invariant under the mapping 
.
Let 
be the number of cyclically
symmetric partitions of 
such that none of 
exceed 
. Let 
be the box containing all integer points 
such that 
, then 
is the number of cyclically
symmetric plane partitions of 
such that 
. Now, let 
be the set of all the orbits
in 
.
Finally, for each point 
in 
, let its height
 |
(1)
|
and for each 
in 
, let 
be the number of points in 
(either 1 or 3) and write
 |
(2)
|
Macdonald (1982) conjectured that
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  | ![product_(i=1)^(m)[(1-q^(3i-1))/(1-q^(3i-2))product_(j=i)^(m)(1-q^(3(m+i+j-1)))/(1-q^(3(2i+j-1)))],](/images/equations/MacdonaldsPlanePartitionConjecture/Inline37.svg) |
(5)
|
(Mills et al. 1982, Macdonald 1995), where the latter form is due to Andrews (1979).
The first few polynomials are
 |  |  |
(6)
| ||
 |  |  |
(7)
| ||
 |  |  |
(8)
| ||
 |  |  |  |  |
(9)
|
which converge coefficientwise to the formal power series with coefficients 1, 1, 0, 0, 1, 0, 0,
2, 1, 0, 2, 1, 0, 4, 3, 0, 5, 4, 0, 8, 8, ..., starting with the constant coefficient.
The coefficients of 
for 
are given by OEIS A096419.
Andrews (1979) proved the 
case, giving the total number of CSPPs
fitting inside an 
box. The general case was proved by Mills et
al. (1982).
