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)¶ Bases:
sage.combinat.diagram_algebras.SubPartitionAlgebra
A Brauer algebra.
The Brauer algebra of rank
is an algebra with basis indexed by the collection of set partitions of
with block size 2.
This algebra is a subalgebra of the partition algebra. For more information, see
PartitionAlgebra
.INPUT:
k
– rank of the algebraq
– the deformation parameter
OPTIONAL ARGUMENTS:
base_ring
– (defaultNone
) a ring containingq
; ifNone
then just takes the parent ofq
prefix
– (default"B"
) a label for the basis elements
EXAMPLES:
We now define the Brauer algebra of rank
with parameter
x
over:
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)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModule
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}}]]
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class
Element
(M, x)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModuleElement
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
ifself
is a basis element. Raises an error ifself
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()
ifDA
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
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
-
class
sage.combinat.diagram_algebras.
PartitionAlgebra
(k, q, base_ring, prefix)¶ Bases:
sage.combinat.diagram_algebras.DiagramAlgebra
A partition algebra.
A partition algebra of rank
over a given ground ring
is an algebra with (
-module) basis indexed by the collection of set partitions of
. Each such set partition can be represented by a graph on nodes
arranged in two rows, with nodes
in the top row from left to right and with nodes
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
. More precisely, for every
, the partition algebra of rank
over
with parameter
is defined to be the
-algebra with basis the collection of all set partitions of
, where the product of two basis elements is given by the rule
where
is the composite set partition obtained by placing the diagram (i.e., graph) of
above the diagram of
, identifying the bottom row nodes of
with the top row nodes of
, and omitting any closed “loops” in the middle. The number
is the number of connected components formed by the omitted loops.
The parameter
is a deformation parameter. Taking
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
(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 algebraq
– the deformation parameter
OPTIONAL ARGUMENTS:
base_ring
– (defaultNone
) a ring containingq
; ifNone
, then Sage automatically chooses the parent ofq
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
with parameter
x
over:
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
:
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)¶ Bases:
sage.combinat.diagram_algebras.SubPartitionAlgebra
A planar algebra.
The planar algebra of rank
is an algebra with basis indexed by the collection of all planar set partitions of
.
This algebra is thus a subalgebra of the partition algebra. For more information, see
PartitionAlgebra
.INPUT:
k
– rank of the algebraq
– the deformation parameter
OPTIONAL ARGUMENTS:
base_ring
– (defaultNone
) a ring containingq
; ifNone
then just takes the parent ofq
prefix
– (default"Pl"
) a label for the basis elements
EXAMPLES:
We define the planar algebra of rank
with parameter
over
:
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
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class
sage.combinat.diagram_algebras.
PropagatingIdeal
(k, q, base_ring, prefix)¶ Bases:
sage.combinat.diagram_algebras.SubPartitionAlgebra
A propagating ideal.
The propagating ideal of rank
is a non-unital algebra with basis indexed by the collection of ideal set partitions of
. We say a set partition is ideal if its propagating number is less than
.
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
with parameter
over
:
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
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class
Element
(M, x)¶ Bases:
sage.combinat.diagram_algebras.DiagramAlgebra.Element
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
-
class
sage.combinat.diagram_algebras.
SubPartitionAlgebra
(k, q, base_ring, prefix, diagrams, category=None)¶ Bases:
sage.combinat.diagram_algebras.DiagramAlgebra
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)¶ Bases:
sage.combinat.diagram_algebras.SubPartitionAlgebra
A Temperley–Lieb algebra.
The Temperley–Lieb algebra of rank
is an algebra with basis indexed by the collection of planar set partitions of
with block size 2.
This algebra is thus a subalgebra of the partition algebra. For more information, see
PartitionAlgebra
.INPUT:
k
– rank of the algebraq
– the deformation parameter
OPTIONAL ARGUMENTS:
base_ring
– (defaultNone
) a ring containingq
; ifNone
then just takes the parent ofq
prefix
– (default"T"
) a label for the basis elements
EXAMPLES:
We define the Temperley–Lieb algebra of rank
with parameter
over
:
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
is a partition diagram of order
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
is a partition diagram of order
with propagating number less than
.
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
EXAMPLES:
sage: import sage.combinat.diagram_algebras as da sage: da.identity_set_partition(2) {{-2, 2}, {-1, 1}}
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sage.combinat.diagram_algebras.
is_planar
(sp)¶ Return
True
if the diagram corresponding to the set partitionsp
is planar; otherwise, returnFalse
.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
andsp2
along with edges joining the bottom row (negative numbers) ofsp1
to the top row (positive numbers) ofsp2
.The vertices of the graph
sp1
appear in the result as pairs(k, 1)
, whereas the vertices of the graphsp2
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
to is a set partition of
. If we have
, then a partition diagram of order
is a set partition of
with
and
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}}]
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sage.combinat.diagram_algebras.
planar_diagrams
(k)¶ Return a list of all planar diagrams of order
k
.A planar diagram of order
is a partition diagram of order
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}}]
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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
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sage.combinat.diagram_algebras.
set_partition_composition
(sp1, sp2)¶ Return a tuple consisting of the composition of the set partitions
sp1
andsp2
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
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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
is a partition diagram of order
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}}]
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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
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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)]
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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
is specified, then the set partition will be a set partition of
. Otherwise,
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