PlanePartitions

Plane partitions of $n$ — 2D arrays of positive integers nonincreasing along rows and down columns, summing to $n$ — as a lazy indexed family.

PlanePartitions(size)the plane partitions summing to size (OEIS A000219).

Typed signatures

Type	Package	Applies
`(integer<0..>) -> indexed_collection<list<list<integer>>>`	`combinatorics`	always

Domain: Collections

Details

A lazy indexed collection with no known simple closed form, unlike ordinary partitions' generating function; the count comes from MacMahon's product $\prod_k (1 - x^k)^{-k}$ — $Count(PlanePartitions(6)) = 48$ .
Each element is the array's rows; entries weakly decrease along every row and down every column, and the whole array sums to $n$ .
Unranked in shape-then-entries order: by row-length shape first, then the flattened array.
Stays TS. A rank is found by walking a table of counts by shape (the layers of a plane partition are nested shapes), then enumerating the one shape it lands in, so At answers up to about $n = 46$ , far past what the whole list could be built. A table over rows and sums would be far larger than the list (BoxedPlanePartitions has one, since a box bounds the rows).

Enumeration

1 more kept as data, checked against the oracles

Basic #

$Count(PlanePartitions(4)) = 13$ , A000219

\mathrm{Count}(\mathrm{PlanePartitions}(4))

Out

13

About

Examples	2
Crosswalk	OEIS SageMath