# plane partitions and all that jazz

wouter meeussen wouter.meeussen at pandora.be
Fri Dec 26 19:38:33 CET 2003

```since it is well known that the plane partitions
are counted by the generating function
GF: Prod(k=1..oo, 1/(1-x^k)^k),

the small step for mankind is:
GF: Prod(k=1..oo, 1/(1-q x^k)^k)

this is the oldest trick in the book,
and yields a triangle of coefficients with
row sums = plane partitions of n = A000219.

Is it true that the successive columns are
col 2= A005993=multidigraphs with loops on 2 nodes with n arcs
col 3= A050531=Multigraphs with loops on 3 nodes with n edges
col 4=(*giant leap*)= ? Multigraphs with loops on 4 nodes with n edges ?

limit value nth column = limit value nth row (backwards)=
Partitions of n objects of 2 colors = A005380

Since the row sums are the plane partitions,
my question is : 'grouped by *what* ?'

Wouter.

ps,

Table[CoefficientList[ Coefficient[
Series[Product[1/(1-q x^k)^k, {k,0,n}],{x,0,n}]
,x^n]/q, q], {n, 7}]

{1},
{2, 1},
{3, 2, 1},
{4, 6, 2, 1},
{5, 10, 6, 2, 1},
{6, 19, 14, 6, 2, 1},
{7, 28, 28, 14, 6, 2, 1}
submission to EIS pending
;-))

```