PartitionAlgebra

$P_n(\delta)$: every set partition of $2n$ points. The largest of the family — all the others are subalgebras cut out by admitting fewer diagrams. Dimension $B(2n)$, the Bell numbers.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPartition algebra↗WikidataQ113101879↗OEISA000110↗SageMathsage.combinat.diagram_algebras.PartitionAlgebra(k, q, base_ring, prefix)✓↗

PartitionAlgebra(n)the partition algebra on $n$ strands

SageMathPartitionAlgebra(n, enumeratio_delta, enumeratio_ring)✓↗OscarEnumeratioDiagramAlgebra(:partition, n)

PlanarPartitionAlgebra(n)only the planar diagrams — dimension $C(2n)$

SageMathPartitionAlgebra(n, enumeratio_delta, enumeratio_ring)✓↗OscarEnumeratioDiagramAlgebra(:partition, n)

Typed signatures

Type	Package	Applies
(integer) -> diagram_algebra	diagram	always

Domain: Diagram algebras

Details

• Dimension $B(2n)$: 2, 15, 203, 4140 for $n = 1,2,3,4$ — the Bell numbers at even index
• Restricting to planar diagrams gives $C(2n)$, the Catalan numbers
• Basis lists the diagrams and AlgebraDimension answers from the closed form, so the dimension is available well past the point where listing is useful
• Element tests membership, which is what makes the inclusions checkable

Examples

Basic #

$B(6)$

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In$\mathrm{AlgebraDimension}(\mathrm{PartitionAlgebra}(3))$Out$203$

Scope #

$B(8)$ — the closed form answers past any useful basis listing

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In$\mathrm{AlgebraDimension}(\mathrm{PartitionAlgebra}(4))$Out$4\,140$

$C(6)$ — planarity cuts 203 down to 132

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In$\mathrm{AlgebraDimension}(\mathrm{PlanarPartitionAlgebra}(3))$Out$132$

Properties #

a block of three is a partition diagram and nothing smaller

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In$\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2, -1\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack, \bigl\lbrack-2\bigr\rbrack\bigr\rbrack)∈\mathrm{PartitionAlgebra}(3)$Out$\mathrm{True}$

About

Examples	4
Checked against	SageMath 4 agreeOscar 3 agree
Crosswalk	WikipediaWikidataOEISSageMath ×2Oscar

Implementation

mappedsagepackages/oracle/src/mappings.ts

Oscar has no diagram algebras; its row checks the dimension against the closed form (oscar/preamble.jl), not against a second implementation.

mappedoscarpackages/oracle/src/mappings.ts

Oscar has no diagram algebras; its row checks the dimension against the closed form (oscar/preamble.jl), not against a second implementation.

See also: Diagram, BrauerAlgebra, TemperleyLiebAlgebra