BoxedPlanePartitions

Plane partitions of any size fitting in an $a \times b \times c$ box — at most $a$ rows, each at most $b$ long, entries at most $c$ — as a lazy indexed family.

SageMathPlanePartitions([a, b, c])↗

BoxedPlanePartitions(a, b, c)the plane partitions fitting an $a \times b \times c$ box.

Typed signatures

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

Domain: Collections

Details

• A lazy indexed collection, exact and closed-form by MacMahon's box formula: $Count(BoxedPlanePartitions(a,b,c)) = \prod_{i=1}^{a} \prod_{j=1}^{b} \prod_{k=1}^{c} \frac{i+j+k-1}{i+j+k-2}$ — $Count(BoxedPlanePartitions(2,2,2)) = 20$.
• Each element is the array's rows, PlanePartitions's ragged carrier: entries weakly decrease along every row and down every column, with trailing zeros trimmed rather than stored.
• Unranked in shape-then-entries order, same as PlanePartitions. A rank is found by walking a table over the rows a box allows (counting what each row starts), so At answers in boxes whose whole list is far too long to build, up to about $(5, 5, 5)$ (267,227,532 partitions); past that a call is refused as too long, and past $2^{53}$ partitions only the count answers.

Enumeration

Examples

1 more kept as data, checked against the oracles

Basic #

$Count(BoxedPlanePartitions(2,2,2)) = 20$, MacMahon's box formula

live ↗

In$\mathrm{Count}(\mathrm{BoxedPlanePartitions}(2, 2, 2))$Out$20$

Scope #

A degenerate $1 \times 1 \times n$ box: $Count = n + 1$

live ↗

In$\mathrm{Count}(\mathrm{BoxedPlanePartitions}(1, 1, 4))$Out$5$

About

Examples	3
Crosswalk	SageMath

See also: PlanePartitions, Count, At