# Diagram and Partition Algebras¶

AUTHORS:

• Mike Hansen (2007): Initial version
• Stephen Doty, Aaron Lauve, George H. Seelinger (2012): Implementation of partition, Brauer, Temperley–Lieb, and ideal partition algebras
class sage.combinat.diagram_algebras.BrauerAlgebra(k, q, base_ring, prefix)

A Brauer algebra.

The Brauer algebra of rank $$k$$ is an algebra with basis indexed by the collection of set partitions of $$\{1, \ldots, k, -1, \ldots, -k\}$$ with block size 2.

This algebra is a subalgebra of the partition algebra. For more information, see PartitionAlgebra.

INPUT:

• k – rank of the algebra
• q – the deformation parameter $$q$$

OPTIONAL ARGUMENTS:

• base_ring – (default None) a ring containing q; if None then just takes the parent of q
• prefix – (default "B") a label for the basis elements

EXAMPLES:

We now define the Brauer algebra of rank $$2$$ with parameter x over $$\ZZ$$:

sage: R.<x> = ZZ[]
sage: B = BrauerAlgebra(2, x, R)
sage: B
Brauer Algebra of rank 2 with parameter x over Univariate Polynomial Ring in x over Integer Ring
sage: B.basis()
Finite family {{{-2, -1}, {1, 2}}: B[{{-2, -1}, {1, 2}}], {{-2, 1}, {-1, 2}}: B[{{-2, 1}, {-1, 2}}], {{-2, 2}, {-1, 1}}: B[{{-2, 2}, {-1, 1}}]}
sage: b = B.basis().list()
sage: b
[B[{{-2, 1}, {-1, 2}}], B[{{-2, 2}, {-1, 1}}], B[{{-2, -1}, {1, 2}}]]
sage: b[2]
B[{{-2, -1}, {1, 2}}]
sage: b[2]^2
x*B[{{-2, -1}, {1, 2}}]
sage: b[2]^5
x^4*B[{{-2, -1}, {1, 2}}]

class sage.combinat.diagram_algebras.DiagramAlgebra(k, q, base_ring, prefix, diagrams, category=None)

Abstract class for diagram algebras and is not designed to be used directly. If used directly, the class could create an “algebra” that is not actually an algebra.

TESTS:

sage: import sage.combinat.diagram_algebras as da
sage: R.<x> = QQ[]
sage: D = da.DiagramAlgebra(2, x, R, 'P', da.partition_diagrams)
sage: sorted(D.basis())
[P[{{-2}, {-1}, {1}, {2}}],
P[{{-2}, {-1}, {1, 2}}],
P[{{-2}, {-1, 1}, {2}}],
P[{{-2}, {-1, 1, 2}}],
P[{{-2}, {-1, 2}, {1}}],
P[{{-2, -1}, {1}, {2}}],
P[{{-2, -1}, {1, 2}}],
P[{{-2, -1, 1}, {2}}],
P[{{-2, -1, 1, 2}}],
P[{{-2, -1, 2}, {1}}],
P[{{-2, 1}, {-1}, {2}}],
P[{{-2, 1}, {-1, 2}}],
P[{{-2, 1, 2}, {-1}}],
P[{{-2, 2}, {-1}, {1}}],
P[{{-2, 2}, {-1, 1}}]]

class Element(M, x)

This subclass provides a few additional methods for partition algebra elements. Most element methods are already implemented elsewhere.

diagram()

Return the underlying diagram of self if self is a basis element. Raises an error if self is not a basis element.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: P = PartitionAlgebra(2, x, R)
sage: elt = 3*P([[1,2],[-2,-1]])
sage: elt.diagram()
{{-2, -1}, {1, 2}}

diagrams()

Return the diagrams in the support of self.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: P = PartitionAlgebra(2, x, R)
sage: elt = 3*P([[1,2],[-2,-1]]) + P([[1,2],[-2], [-1]])
sage: elt.diagrams()
[{{-2}, {-1}, {1, 2}}, {{-2, -1}, {1, 2}}]

DiagramAlgebra.one_basis()

The following constructs the identity element of the diagram algebra.

It is not called directly; instead one should use DA.one() if DA is a defined diagram algebra.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: R.<x> = QQ[]
sage: D = da.DiagramAlgebra(2, x, R, 'P', da.partition_diagrams)
sage: D.one_basis()
{{-2, 2}, {-1, 1}}

DiagramAlgebra.order()

Return the order of self.

The order of a partition algebra is defined as half of the number of nodes in the diagrams.

EXAMPLES:

sage: q = var('q')
sage: PA = PartitionAlgebra(2, q)
sage: PA.order()
2

DiagramAlgebra.product_on_basis(d1, d2)

Returns the product $$D_{d_1} D_{d_2}$$ by two basis diagrams.

TESTS:

sage: import sage.combinat.diagram_algebras as da
sage: R.<x> = QQ[]
sage: D = da.DiagramAlgebra(2, x, R, 'P', da.partition_diagrams)
sage: sp = SetPartition([[1,2],[-1,-2]])
sage: D.product_on_basis(sp, sp)
x*P[{{-2, -1}, {1, 2}}]

DiagramAlgebra.set_partitions()

Return the collection of underlying set partitions indexing the basis elements of a given diagram algebra.

TESTS:

sage: import sage.combinat.diagram_algebras as da
sage: R.<x> = QQ[]
sage: D = da.DiagramAlgebra(2, x, R, 'P', da.partition_diagrams)
sage: list(D.set_partitions()) == da.partition_diagrams(2)
True

class sage.combinat.diagram_algebras.PartitionAlgebra(k, q, base_ring, prefix)

A partition algebra.

A partition algebra of rank $$k$$ over a given ground ring $$R$$ is an algebra with ($$R$$-module) basis indexed by the collection of set partitions of $$\{1, \ldots, k, -1, \ldots, -k\}$$. Each such set partition can be represented by a graph on nodes $$\{1, \ldots, k, -1, \ldots, -k\}$$ arranged in two rows, with nodes $$1, \dots, k$$ in the top row from left to right and with nodes $$-1, \ldots, -k$$ in the bottom row from left to right, and edges drawn such that the connected components of the graph are precisely the parts of the set partition. (This choice of edges is often not unique, and so there are often many graphs representing one and the same set partition; the representation nevertheless is useful and vivid. We often speak of “diagrams” to mean graphs up to such equivalence of choices of edges; of course, we could just as well speak of set partitions.)

There is not just one partition algebra of given rank over a given ground ring, but rather a whole family of them, indexed by the elements of $$R$$. More precisely, for every $$q \in R$$, the partition algebra of rank $$k$$ over $$R$$ with parameter $$q$$ is defined to be the $$R$$-algebra with basis the collection of all set partitions of $$\{1, \ldots, k, -1, \ldots, -k\}$$, where the product of two basis elements is given by the rule

$a \cdot b = q^N (a \circ b),$

where $$a \circ b$$ is the composite set partition obtained by placing the diagram (i.e., graph) of $$a$$ above the diagram of $$b$$, identifying the bottom row nodes of $$a$$ with the top row nodes of $$b$$, and omitting any closed “loops” in the middle. The number $$N$$ is the number of connected components formed by the omitted loops.

The parameter $$q$$ is a deformation parameter. Taking $$q = 1$$ produces the semigroup algebra (over the base ring) of the partition monoid, in which the product of two set partitions is simply given by their composition.

The Iwahori–Hecke algebra of type $$A$$ (with a single parameter) is naturally a subalgebra of the partition algebra.

The partition algebra is regarded as an example of a “diagram algebra” due to the fact that its natural basis is given by certain graphs often called diagrams.

An excellent reference for partition algebras and their various subalgebras (Brauer algebra, Temperley–Lieb algebra, etc) is the paper [HR2005].

INPUT:

• k – rank of the algebra
• q – the deformation parameter $$q$$

OPTIONAL ARGUMENTS:

• base_ring – (default None) a ring containing q; if None, then Sage automatically chooses the parent of q
• prefix – (default "P") a label for the basis elements

EXAMPLES:

The following shorthand simultaneously defines the univariate polynomial ring over the rationals as well as the variable x:

sage: R.<x> = PolynomialRing(QQ)
sage: R
Univariate Polynomial Ring in x over Rational Field
sage: x
x
sage: x.parent() is R
True


We now define the partition algebra of rank $$2$$ with parameter x over $$\ZZ$$:

sage: R.<x> = ZZ[]
sage: P = PartitionAlgebra(2, x, R)
sage: P
Partition Algebra of rank 2 with parameter x over Univariate Polynomial Ring in x over Integer Ring
sage: P.basis().list()
[P[{{-2, -1, 1, 2}}], P[{{-2, -1, 2}, {1}}],
P[{{-2, -1, 1}, {2}}], P[{{-2}, {-1, 1, 2}}],
P[{{-2, 1, 2}, {-1}}], P[{{-2, 1}, {-1, 2}}],
P[{{-2, 2}, {-1, 1}}], P[{{-2, -1}, {1, 2}}],
P[{{-2, -1}, {1}, {2}}], P[{{-2}, {-1, 2}, {1}}],
P[{{-2, 2}, {-1}, {1}}], P[{{-2}, {-1, 1}, {2}}],
P[{{-2, 1}, {-1}, {2}}], P[{{-2}, {-1}, {1, 2}}],
P[{{-2}, {-1}, {1}, {2}}]]
sage: E = P([[1,2],[-2,-1]]); E
P[{{-2, -1}, {1, 2}}]
sage: E in P.basis()
True
sage: E^2
x*P[{{-2, -1}, {1, 2}}]
sage: E^5
x^4*P[{{-2, -1}, {1, 2}}]
sage: (P([[2,-2],[-1,1]]) - 2*P([[1,2],[-1,-2]]))^2
(4*x-4)*P[{{-2, -1}, {1, 2}}] + P[{{-2, 2}, {-1, 1}}]


One can work with partition algebras using a symbol for the parameter, leaving the base ring unspecified. This implies that the underlying base ring is Sage’s symbolic ring.

sage: q = var('q')
sage: PA = PartitionAlgebra(2, q); PA
Partition Algebra of rank 2 with parameter q over Symbolic Ring
sage: PA([[1,2],[-2,-1]])^2 == q*PA([[1,2],[-2,-1]])
True
sage: (PA([[2, -2], [1, -1]]) - 2*PA([[-2, -1], [1, 2]]))^2 == (4*q-4)*PA([[1, 2], [-2, -1]]) + PA([[2, -2], [1, -1]])
True


The identity element of the partition algebra is the set partition $$\{\{1,-1\}, \{2,-2\}, \ldots, \{k,-k\}\}$$:

sage: P = PA.basis().list()
sage: PA.one()
P[{{-2, 2}, {-1, 1}}]
sage: PA.one()*P[7] == P[7]
True
sage: P[7]*PA.one() == P[7]
True


We now give some further examples of the use of the other arguments. One may wish to “specialize” the parameter to a chosen element of the base ring:

sage: R.<q> = RR[]
sage: PA = PartitionAlgebra(2, q, R, prefix='B')
sage: PA
Partition Algebra of rank 2 with parameter q over
Univariate Polynomial Ring in q over Real Field with 53 bits of precision
sage: PA([[1,2],[-1,-2]])
1.00000000000000*B[{{-2, -1}, {1, 2}}]
sage: PA = PartitionAlgebra(2, 5, base_ring=ZZ, prefix='B')
sage: PA
Partition Algebra of rank 2 with parameter 5 over Integer Ring
sage: (PA([[2, -2], [1, -1]]) - 2*PA([[-2, -1], [1, 2]]))^2 == 16*PA([[-2, -1], [1, 2]]) + PA([[2, -2], [1, -1]])
True


TESTS:

A computation that returned an incorrect result until trac ticket #15958:

sage: A = PartitionAlgebra(1,17)
sage: g = SetPartitionsAk(1).list()
sage: a = A[g[1]]
sage: a
P[{{-1}, {1}}]
sage: a*a
17*P[{{-1}, {1}}]


REFERENCES:

 [HR2005] (1, 2) Tom Halverson and Arun Ram, Partition algebras. European Journal of Combinatorics 26 (2005), 869–921.
class sage.combinat.diagram_algebras.PlanarAlgebra(k, q, base_ring, prefix)

A planar algebra.

The planar algebra of rank $$k$$ is an algebra with basis indexed by the collection of all planar set partitions of $$\{1, \ldots, k, -1, \ldots, -k\}$$.

This algebra is thus a subalgebra of the partition algebra. For more information, see PartitionAlgebra.

INPUT:

• k – rank of the algebra
• q – the deformation parameter $$q$$

OPTIONAL ARGUMENTS:

• base_ring – (default None) a ring containing q; if None then just takes the parent of q
• prefix – (default "Pl") a label for the basis elements

EXAMPLES:

We define the planar algebra of rank $$2$$ with parameter $$x$$ over $$\ZZ$$:

sage: R.<x> = ZZ[]
sage: Pl = PlanarAlgebra(2, x, R); Pl
Planar Algebra of rank 2 with parameter x over Univariate Polynomial Ring in x over Integer Ring
sage: Pl.basis().list()
[Pl[{{-2, -1, 1, 2}}], Pl[{{-2, -1, 2}, {1}}],
Pl[{{-2, -1, 1}, {2}}], Pl[{{-2}, {-1, 1, 2}}],
Pl[{{-2, 1, 2}, {-1}}], Pl[{{-2, 2}, {-1, 1}}],
Pl[{{-2, -1}, {1, 2}}], Pl[{{-2, -1}, {1}, {2}}],
Pl[{{-2}, {-1, 2}, {1}}], Pl[{{-2, 2}, {-1}, {1}}],
Pl[{{-2}, {-1, 1}, {2}}], Pl[{{-2, 1}, {-1}, {2}}],
Pl[{{-2}, {-1}, {1, 2}}], Pl[{{-2}, {-1}, {1}, {2}}]]
sage: E = Pl([[1,2],[-1,-2]])
sage: E^2 == x*E
True
sage: E^5 == x^4*E
True

class sage.combinat.diagram_algebras.PropagatingIdeal(k, q, base_ring, prefix)

A propagating ideal.

The propagating ideal of rank $$k$$ is a non-unital algebra with basis indexed by the collection of ideal set partitions of $$\{1, \ldots, k, -1, \ldots, -k\}$$. We say a set partition is ideal if its propagating number is less than $$k$$.

This algebra is a non-unital subalgebra and an ideal of the partition algebra. For more information, see PartitionAlgebra.

EXAMPLES:

We now define the propagating ideal of rank $$2$$ with parameter $$x$$ over $$\ZZ$$:

sage: R.<x> = QQ[]
sage: I = PropagatingIdeal(2, x, R); I
Propagating Ideal of rank 2 with parameter x over Univariate Polynomial Ring in x over Rational Field
sage: I.basis().list()
[I[{{-2, -1, 1, 2}}], I[{{-2, -1, 2}, {1}}],
I[{{-2, -1, 1}, {2}}], I[{{-2}, {-1, 1, 2}}],
I[{{-2, 1, 2}, {-1}}], I[{{-2, -1}, {1, 2}}],
I[{{-2, -1}, {1}, {2}}], I[{{-2}, {-1, 2}, {1}}],
I[{{-2, 2}, {-1}, {1}}], I[{{-2}, {-1, 1}, {2}}],
I[{{-2, 1}, {-1}, {2}}], I[{{-2}, {-1}, {1, 2}}],
I[{{-2}, {-1}, {1}, {2}}]]
sage: E = I([[1,2],[-1,-2]])
sage: E^2 == x*E
True
sage: E^5 == x^4*E
True

class Element(M, x)

Need to take care of exponents since we are not unital.

PropagatingIdeal.one_basis()

The propagating ideal is a non-unital algebra, i.e. it does not have a multiplicative identity.

EXAMPLES:

sage: R.<q> = QQ[]
sage: I = PropagatingIdeal(2, q, R)
sage: I.one_basis()
Traceback (most recent call last):
...
ValueError: The ideal partition algebra is not unital
sage: I.one()
Traceback (most recent call last):
...
ValueError: The ideal partition algebra is not unital

class sage.combinat.diagram_algebras.SubPartitionAlgebra(k, q, base_ring, prefix, diagrams, category=None)

A subalgebra of the partition algebra indexed by a subset of the diagrams.

ambient()

Return the partition algebra self is a sub-algebra of. Generally, this method is not called directly.

EXAMPLES:

sage: x = var('x')
sage: BA = BrauerAlgebra(2, x)
sage: BA.ambient()
Partition Algebra of rank 2 with parameter x over Symbolic Ring

lift(x)

Lift a diagram subalgebra element to the corresponding element in the ambient space. This method is not intended to be called directly.

EXAMPLES:

sage: R.<x> = QQ[]
sage: BA = BrauerAlgebra(2, x, R)
sage: E = BA([[1,2],[-1,-2]])
sage: lifted = BA.lift(E); lifted
B[{{-2, -1}, {1, 2}}]
sage: lifted.parent() is BA.ambient()
True

retract(x)

Retract an appropriate partition algebra element to the corresponding element in the partition subalgebra. This method is not intended to be called directly.

EXAMPLES:

sage: R.<x> = QQ[]
sage: BA = BrauerAlgebra(2, x, R)
sage: PA = BA.ambient()
sage: E = PA([[1,2], [-1,-2]])
sage: BA.retract(E) in BA
True

class sage.combinat.diagram_algebras.TemperleyLiebAlgebra(k, q, base_ring, prefix)

A Temperley–Lieb algebra.

The Temperley–Lieb algebra of rank $$k$$ is an algebra with basis indexed by the collection of planar set partitions of $$\{1, \ldots, k, -1, \ldots, -k\}$$ with block size 2.

This algebra is thus a subalgebra of the partition algebra. For more information, see PartitionAlgebra.

INPUT:

• k – rank of the algebra
• q – the deformation parameter $$q$$

OPTIONAL ARGUMENTS:

• base_ring – (default None) a ring containing q; if None then just takes the parent of q
• prefix – (default "T") a label for the basis elements

EXAMPLES:

We define the Temperley–Lieb algebra of rank $$2$$ with parameter $$x$$ over $$\ZZ$$:

sage: R.<x> = ZZ[]
sage: T = TemperleyLiebAlgebra(2, x, R); T
Temperley-Lieb Algebra of rank 2 with parameter x over Univariate Polynomial Ring in x over Integer Ring
sage: T.basis()
Finite family {{{-2, 2}, {-1, 1}}: T[{{-2, 2}, {-1, 1}}], {{-2, -1}, {1, 2}}: T[{{-2, -1}, {1, 2}}]}
sage: b = T.basis().list()
sage: b
[T[{{-2, 2}, {-1, 1}}], T[{{-2, -1}, {1, 2}}]]
sage: b[1]
T[{{-2, -1}, {1, 2}}]
sage: b[1]^2 == x*b[1]
True
sage: b[1]^5 == x^4*b[1]
True

sage.combinat.diagram_algebras.brauer_diagrams(k)

Return a list of all Brauer diagrams of order k.

A Brauer diagram of order $$k$$ is a partition diagram of order $$k$$ with block size 2.

INPUT:

• k – the order of the Brauer diagrams

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.brauer_diagrams(2)
[{{-2, 1}, {-1, 2}}, {{-2, 2}, {-1, 1}}, {{-2, -1}, {1, 2}}]
sage: da.brauer_diagrams(5/2)
[{{-3, 3}, {-2, 1}, {-1, 2}}, {{-3, 3}, {-2, 2}, {-1, 1}}, {{-3, 3}, {-2, -1}, {1, 2}}]

sage.combinat.diagram_algebras.ideal_diagrams(k)

Return a list of all “ideal” diagrams of order k.

An ideal diagram of order $$k$$ is a partition diagram of order $$k$$ with propagating number less than $$k$$.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.ideal_diagrams(2)
[{{-2, -1, 1, 2}}, {{-2, -1, 2}, {1}}, {{-2, -1, 1}, {2}}, {{-2}, {-1, 1, 2}},
{{-2, 1, 2}, {-1}}, {{-2, -1}, {1, 2}}, {{-2, -1}, {1}, {2}},
{{-2}, {-1, 2}, {1}}, {{-2, 2}, {-1}, {1}}, {{-2}, {-1, 1}, {2}}, {{-2, 1},
{-1}, {2}}, {{-2}, {-1}, {1, 2}}, {{-2}, {-1}, {1}, {2}}]
sage: da.ideal_diagrams(3/2)
[{{-2, -1, 1, 2}}, {{-2, -1, 2}, {1}}, {{-2, 1, 2}, {-1}}, {{-2, 2}, {-1}, {1}}]

sage.combinat.diagram_algebras.identity_set_partition(k)

Return the identity set partition $$\{\{1, -1\}, \ldots, \{k, -k\}\}$$

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.identity_set_partition(2)
{{-2, 2}, {-1, 1}}

sage.combinat.diagram_algebras.is_planar(sp)

Return True if the diagram corresponding to the set partition sp is planar; otherwise, return False.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.is_planar( da.to_set_partition([[1,-2],[2,-1]]))
False
sage: da.is_planar( da.to_set_partition([[1,-1],[2,-2]]))
True

sage.combinat.diagram_algebras.pair_to_graph(sp1, sp2)

Return a graph consisting of the disjoint union of the graphs of set partitions sp1 and sp2 along with edges joining the bottom row (negative numbers) of sp1 to the top row (positive numbers) of sp2.

The vertices of the graph sp1 appear in the result as pairs (k, 1), whereas the vertices of the graph sp2 appear as pairs (k, 2).

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: sp1 = da.to_set_partition([[1,-2],[2,-1]])
sage: sp2 = da.to_set_partition([[1,-2],[2,-1]])
sage: g = da.pair_to_graph( sp1, sp2 ); g
Graph on 8 vertices

sage: g.vertices()
[(-2, 1), (-2, 2), (-1, 1), (-1, 2), (1, 1), (1, 2), (2, 1), (2, 2)]
sage: g.edges()
[((-2, 1), (1, 1), None), ((-2, 1), (2, 2), None),
((-2, 2), (1, 2), None), ((-1, 1), (1, 2), None),
((-1, 1), (2, 1), None), ((-1, 2), (2, 2), None)]


Another example which used to be wrong until trac ticket #15958:

sage: sp3 = da.to_set_partition([[1, -1], [2], [-2]])
sage: sp4 = da.to_set_partition([[1], [-1], [2], [-2]])
sage: g = da.pair_to_graph( sp3, sp4 ); g
Graph on 8 vertices

sage: g.vertices()
[(-2, 1), (-2, 2), (-1, 1), (-1, 2), (1, 1), (1, 2), (2, 1), (2, 2)]
sage: g.edges()
[((-2, 1), (2, 2), None), ((-1, 1), (1, 1), None),
((-1, 1), (1, 2), None)]

sage.combinat.diagram_algebras.partition_diagrams(k)

Return a list of all partition diagrams of order k.

A partition diagram of order $$k \in \ZZ$$ to is a set partition of $$\{1, \dots, k, -1, \ldots, -k\}$$. If we have $$k - 1/2 \in ZZ$$, then a partition diagram of order $$k \in 1/2 \ZZ$$ is a set partition of $$\{1, \ldots, k+1/2, -1, \ldots, -(k+1/2)\}$$ with $$k+1/2$$ and $$-(k+1/2)$$ in the same block. See [HR2005].

INPUT:

• k – the order of the partition diagrams

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.partition_diagrams(2)
[{{-2, -1, 1, 2}}, {{-2, -1, 2}, {1}}, {{-2, -1, 1}, {2}},
{{-2}, {-1, 1, 2}}, {{-2, 1, 2}, {-1}}, {{-2, 1}, {-1, 2}},
{{-2, 2}, {-1, 1}}, {{-2, -1}, {1, 2}}, {{-2, -1}, {1}, {2}},
{{-2}, {-1, 2}, {1}}, {{-2, 2}, {-1}, {1}}, {{-2}, {-1, 1}, {2}},
{{-2, 1}, {-1}, {2}}, {{-2}, {-1}, {1, 2}}, {{-2}, {-1}, {1}, {2}}]
sage: da.partition_diagrams(3/2)
[{{-2, -1, 1, 2}}, {{-2, -1, 2}, {1}}, {{-2, 2}, {-1, 1}},
{{-2, 1, 2}, {-1}}, {{-2, 2}, {-1}, {1}}]

sage.combinat.diagram_algebras.planar_diagrams(k)

Return a list of all planar diagrams of order k.

A planar diagram of order $$k$$ is a partition diagram of order $$k$$ that has no crossings.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.planar_diagrams(2)
[{{-2, -1, 1, 2}}, {{-2, -1, 2}, {1}}, {{-2, -1, 1}, {2}},
{{-2}, {-1, 1, 2}}, {{-2, 1, 2}, {-1}}, {{-2, 2}, {-1, 1}},
{{-2, -1}, {1, 2}}, {{-2, -1}, {1}, {2}}, {{-2}, {-1, 2}, {1}},
{{-2, 2}, {-1}, {1}}, {{-2}, {-1, 1}, {2}}, {{-2, 1}, {-1}, {2}},
{{-2}, {-1}, {1, 2}}, {{-2}, {-1}, {1}, {2}}]
sage: da.planar_diagrams(3/2)
[{{-2, -1, 1, 2}}, {{-2, -1, 2}, {1}}, {{-2, 2}, {-1, 1}},
{{-2, 1, 2}, {-1}}, {{-2, 2}, {-1}, {1}}]

sage.combinat.diagram_algebras.propagating_number(sp)

Return the propagating number of the set partition sp.

The propagating number is the number of blocks with both a positive and negative number.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: sp1 = da.to_set_partition([[1,-2],[2,-1]])
sage: sp2 = da.to_set_partition([[1,2],[-2,-1]])
sage: da.propagating_number(sp1)
2
sage: da.propagating_number(sp2)
0

sage.combinat.diagram_algebras.set_partition_composition(sp1, sp2)

Return a tuple consisting of the composition of the set partitions sp1 and sp2 and the number of components removed from the middle rows of the graph.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: sp1 = da.to_set_partition([[1,-2],[2,-1]])
sage: sp2 = da.to_set_partition([[1,-2],[2,-1]])
sage: da.set_partition_composition(sp1, sp2) == (da.identity_set_partition(2), 0)
True

sage.combinat.diagram_algebras.temperley_lieb_diagrams(k)

Return a list of all Temperley–Lieb diagrams of order k.

A Temperley–Lieb diagram of order $$k$$ is a partition diagram of order $$k$$ with block size 2 and is planar.

INPUT:

• k – the order of the Temperley–Lieb diagrams

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.temperley_lieb_diagrams(2)
[{{-2, 2}, {-1, 1}}, {{-2, -1}, {1, 2}}]
sage: da.temperley_lieb_diagrams(5/2)
[{{-3, 3}, {-2, 2}, {-1, 1}}, {{-3, 3}, {-2, -1}, {1, 2}}]

sage.combinat.diagram_algebras.to_Brauer_partition(l, k=None)

Same as to_set_partition() but assumes omitted elements are connected straight through.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.to_Brauer_partition([[1,2],[-1,-2]]) == SetPartition([[1,2],[-1,-2]])
True
sage: da.to_Brauer_partition([[1,3],[-1,-3]]) == SetPartition([[1,3],[-3,-1],[2,-2]])
True
sage: da.to_Brauer_partition([[1,2],[-1,-2]], k=4) == SetPartition([[1,2],[-1,-2],[3,-3],[4,-4]])
True
sage: da.to_Brauer_partition([[1,-4],[-3,-1],[3,4]]) == SetPartition([[-3,-1],[2,-2],[1,-4],[3,4]])
True

sage.combinat.diagram_algebras.to_graph(sp)

Return a graph representing the set partition sp.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: g = da.to_graph( da.to_set_partition([[1,-2],[2,-1]])); g
Graph on 4 vertices

sage: g.vertices()
[-2, -1, 1, 2]
sage: g.edges()
[(-2, 1, None), (-1, 2, None)]

sage.combinat.diagram_algebras.to_set_partition(l, k=None)

Convert a list of a list of numbers to a set partitions. Each list of numbers in the outer list specifies the numbers contained in one of the blocks in the set partition.

If $$k$$ is specified, then the set partition will be a set partition of $$\{1, \ldots, k, -1, \ldots, -k\}$$. Otherwise, $$k$$ will default to the minimum number needed to contain all of the specified numbers.

EXAMPLES:

sage: import sage.combinat.diagram_algebras as da
sage: da.to_set_partition([[1,-1],[2,-2]]) == da.identity_set_partition(2)
True


Descent Algebras

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Symmetric Group Algebra