plane partitions and all that jazz

Fri Dec 26 22:58:30 CET 2003

```Coefficient of q^m*x^n in expansion of Prod(k=1..oo, 1/(1-q x^k)^k)
is the number of planar partitions of n with trace m,
cf. G. E. Andrews, The Theory of Partitions,
Ch. XI, exercise 5 and Ch. XII, exercise 5.

Vl.

____________

> 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
> ;-))
>
>
>
>

```