# some more dissections of plane partitions

wouter meeussen wouter.meeussen at pandora.be
Sun Dec 28 14:25:24 CET 2003

```Table[CoefficientList[ Coefficient[
Series[Product[Product[ 1/(1 - q^j x^k), {j, k}], {k, 0, n}],
{x, 0, n}], x^n]/q, q], {n, 12}]

{ 1},
{ 1, 2},
{ 1, 2, 3},
{ 1, 3, 4, 5},
{ 1, 3, 6, 7, 7},
{ 1, 4, 8, 12, 12, 11},
{ 1, 4, 10, 16, 21, 19, 15},
{ 1, 5, 12, 23, 31, 36, 30, 22},
{ 1, 5, 15, 28, 45, 55, 58, 45, 30},
{ 1, 6, 17, 37, 60, 84, 94, 92, 67, 42},
{ 1, 6, 20, 44, 80, 115, 147, 153, 140, 97, 56},
{ 1, 7, 23, 55, 101, 161, 211, 249, 244, 211, 139, 77}

T[n,n] are the partitions numbers A000041.
Rowsum is by definition A000219 (plane partitions)

my best guess is that it counts the plane partitions of n
with k integers :

1 integer : {{4}}
2 integers: {{3,1}} & {{2,2}} & {{3},{1}} & {{2},{2}}
3 integers: {{2,1,1}} & {{2},{1},{1}} & {{2,1},{1}}
4 integers: {{1,1,1,1}} & {{1},{1},{1},{1}} & {{1,1},{1,1}} &
{{1,1,1},{1}} & {{1,1},{1},{1}}

but this counts as 1,4,3,5 instead of 1,3,4,5.

Any better ideas?
Any literature lying around like Vladeta's
"G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976
(Ch. XI, exercise 5 and Ch. XII, exercise 5)".

W.

```