SkewPartitions

Reduced skew shapes $\lambda/\mu$ with $n$ cells — no empty row or column — as a lazy indexed family.

SageMathSkewPartitions(n)↗

SkewPartitions(size)the reduced skew shapes $\lambda/\mu$ with size cells total ($|\lambda| - |\mu| = n$).

Typed signatures

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

Domain: Collections

Details

• A lazy indexed collection with no closed form, following Sage's SkewPartitions(n): the count is a table over the last row of a shape and the cells left (exact, until it passes $2^{53}$ at $n = 34$, where the count is unknown), and a place in the order is found by walking that table, so At is quick at sizes whose whole list is far too long.
• Each element packs both partitions as [λ, μ]; "reduced" means every row of $\lambda$ strictly exceeds the matching row of $\mu$ (no empty row) and every column $1..\lambda_1$ is covered by some row's cells (no empty column). $\mu$ has no zero parts: [[2, 1], [1, 0]] is not an element, [[2, 1], [1]] is.
• Unranked lexicographically, by $\lambda$ first, then by $\mu$.

Enumeration

Examples

1 more kept as data, checked against the oracles

Basic #

$Count(SkewPartitions(4)) = 28$: pairs $(\lambda, \mu)$ with $\mu \subseteq \lambda$ and $|\lambda| - |\mu| = 4$

live ↗

In$\mathrm{Count}(\mathrm{SkewPartitions}(4))$Out$28$

Scope #

The first skew shape of size 3, as an outer/inner partition pair

live ↗

In$\mathrm{SkewPartitions}(3)[1]$Out$\bigl\lbrack\bigl\lbrack1, 1, 1\bigr\rbrack, \bigl\lbrack\bigr\rbrack\bigr\rbrack$

About

Examples	3
Crosswalk	SageMath

See also: IntegerPartitions, Count, At