MotzkinAlgebra

$M_n(\delta)$: the planar diagrams whose blocks have size at most two — so points may also be left unpaired. Dimension $M(2n)$, the Motzkin numbers.

WikipediaMotzkin number↗MathWorldMotzkinNumber↗WikidataQ2915234↗DLMF26.6.E2↗OEISA001006↗

MotzkinAlgebra(n)the Motzkin algebra on $n$ strands

OscarEnumeratioDiagramAlgebra(:motzkin, n)

RookAlgebra(n)partial permutations — blocks of size ≤ 2, each pairing a top with a bottom

OscarEnumeratioDiagramAlgebra(:motzkin, n)

SymmetricGroupAlgebra(n)permutation diagrams only — dimension $n!$

OscarEnumeratioDiagramAlgebra(:motzkin, n)

Typed signatures

Type	Package	Applies
(integer) -> diagram_algebra	diagram	always

Domain: Diagram algebras

Details

• Dimension $M(2n)$ = 2, 9, 51, 323 — the Motzkin numbers, which count the same paths with a flat step that Catalan counts without
• Contains TemperleyLiebAlgebra: dropping the requirement that every point be paired is exactly what turns Catalan into Motzkin
• RookAlgebra(n) counts $\sum_k \binom{n}{k}^2 k!$ — the partial permutations, i.e. the placements of non-attacking rooks
• SymmetricGroupAlgebra(n) is the group algebra of $S_n$, the diagrams that are honest bijections

Examples

Basic #

$M(6)$

live ↗

In$\mathrm{AlgebraDimension}(\mathrm{MotzkinAlgebra}(3))$Out$51$

live ↗

In$\mathrm{AlgebraDimension}(\mathrm{RookAlgebra}(3))$Out$34$

$5!$

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In$\mathrm{AlgebraDimension}(\mathrm{SymmetricGroupAlgebra}(5))$Out$120$

Properties #

a cup-cap is not a permutation

live ↗

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

About

Examples	4
Checked against	Oscar 3 agreeSageMath 2 agree
Crosswalk	WikipediaMathWorldWikidataDLMFOEISOscar

Implementation

mappedoscarpackages/oracle/src/mappings.ts

Sage has no Motzkin algebra.

See also: TemperleyLiebAlgebra, PartitionAlgebra, Diagram