Set Partitions¶
AUTHORS:
- Mike Hansen
- MuPAD-Combinat developers (for algorithms and design inspiration).
- Travis Scrimshaw (2013-02-28): Removed
CombinatorialClass
and added entry point throughSetPartition
.
-
class
sage.combinat.set_partition.
SetPartition
(parent, s)¶ Bases:
sage.structure.list_clone.ClonableArray
A partition of a set.
A set partition \(p\) of a set \(S\) is a partition of \(S\) into subsets called parts and represented as a set of sets. By extension, a set partition of a nonnegative integer \(n\) is the set partition of the integers from 1 to \(n\). The number of set partitions of \(n\) is called the \(n\)-th Bell number.
There is a natural integer partition associated with a set partition, namely the nonincreasing sequence of sizes of all its parts.
There is a classical lattice associated with all set partitions of \(n\). The infimum of two set partitions is the set partition obtained by intersecting all the parts of both set partitions. The supremum is obtained by transitive closure of the relation \(i\) related to \(j\) if and only if they are in the same part in at least one of the set partitions.
We will use terminology from partitions, in particular the length of a set partition \(A = \{A_1, \ldots, A_k\}\) is the number of parts of \(A\) and is denoted by \(|A| := k\). The size of \(A\) is the cardinality of \(S\). We will also sometimes use the notation \([n] := \{1, 2, \ldots, n\}\).
EXAMPLES:
There are 5 set partitions of the set \(\{1,2,3\}\):
sage: SetPartitions(3).cardinality() 5
Here is the list of them:
sage: SetPartitions(3).list() [{{1, 2, 3}}, {{1}, {2, 3}}, {{1, 3}, {2}}, {{1, 2}, {3}}, {{1}, {2}, {3}}]
There are 6 set partitions of \(\{1,2,3,4\}\) whose underlying partition is \([2, 1, 1]\):
sage: SetPartitions(4, [2,1,1]).list() [{{1}, {2}, {3, 4}}, {{1}, {2, 4}, {3}}, {{1}, {2, 3}, {4}}, {{1, 4}, {2}, {3}}, {{1, 3}, {2}, {4}}, {{1, 2}, {3}, {4}}]
Since trac ticket #14140, we can create a set partition directly by
SetPartition
, which creates the base set by taking the union of the parts passed in:sage: s = SetPartition([[1,3],[2,4]]); s {{1, 3}, {2, 4}} sage: s.parent() Set partitions
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apply_permutation
(p)¶ Apply
p
to the underlying set ofself
.INPUT:
p
– A permutation
EXAMPLES:
sage: x = SetPartition([[1,2], [3,5,4]]) sage: p = Permutation([2,1,4,5,3]) sage: x.apply_permutation(p) {{1, 2}, {3, 4, 5}} sage: q = Permutation([3,2,1,5,4]) sage: x.apply_permutation(q) {{1, 4, 5}, {2, 3}}
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arcs
()¶ Return
self
as a list of arcs.Consider a set partition \(X = \{X_1, X_2, \ldots, X_n\}\), then the arcs are the pairs \(i - j\) such that \(i,j \in X_k\) for some \(k\).
EXAMPLES:
sage: A = SetPartition([[1],[2,3],[4]]) sage: A.arcs() [(2, 3)] sage: B = SetPartition([[1,3,6,7],[2,5],[4]]) sage: B.arcs() [(1, 3), (3, 6), (6, 7), (2, 5)]
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base_set
()¶ Return the base set of
self
, which is the union of all parts ofself
.EXAMPLES:
sage: SetPartition([[1], [2,3], [4]]).base_set() {1, 2, 3, 4} sage: SetPartition([[1,2,3,4]]).base_set() {1, 2, 3, 4} sage: SetPartition([]).base_set() {}
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base_set_cardinality
()¶ Return the cardinality of the base set of
self
, which is the sum of the sizes of the parts ofself
.This is also known as the size (sometimes the weight) of a set partition.
EXAMPLES:
sage: SetPartition([[1], [2,3], [4]]).base_set_cardinality() 4 sage: SetPartition([[1,2,3,4]]).base_set_cardinality() 4
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cardinality
()¶ Returns the len of
self
EXAMPLES:
sage: from sage.structure.list_clone_demo import IncreasingArrays sage: len(IncreasingArrays()([1,2,3])) 3
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check
()¶ Check that we are a valid ordered set partition.
EXAMPLES:
sage: OS = OrderedSetPartitions(4) sage: s = OS([[1, 3], [2, 4]]) sage: s.check()
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coarsenings
()¶ Return a list of coarsenings of
self
.EXAMPLES:
sage: SetPartition([[1,3],[2,4]]).coarsenings() [{{1, 2, 3, 4}}, {{1, 3}, {2, 4}}] sage: SetPartition([[1],[2,4],[3]]).coarsenings() [{{1, 2, 3, 4}}, {{1}, {2, 3, 4}}, {{1, 3}, {2, 4}}, {{1, 2, 4}, {3}}, {{1}, {2, 4}, {3}}] sage: SetPartition([]).coarsenings() [{}]
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inf
(other)¶ The product of the set partitions
self
andother
.The product of two set partitions \(B\) and \(C\) is defined as the set partition whose parts are the nonempty intersections between each part of \(B\) and each part of \(C\). This product is also the infimum of \(B\) and \(C\) in the classical set partition lattice (that is, the coarsest set partition which is finer than each of \(B\) and \(C\)). Consequently,
inf
acts as an alias for this method.See also
EXAMPLES:
sage: x = SetPartition([ [1,2], [3,5,4] ]) sage: y = SetPartition(( (3,1,2), (5,4) )) sage: x * y {{1, 2}, {3}, {4, 5}} sage: S = SetPartitions(4) sage: sp1 = S([[2,3,4], [1]]) sage: sp2 = S([[1,3], [2,4]]) sage: s = S([[2,4], [3], [1]]) sage: sp1.inf(sp2) == s True
TESTS:
Here is a different implementation of the
__mul__
method (one that was formerly used for theinf
method, before it was realized that the methods do the same thing):sage: def mul2(s, t): ....: temp = [ss.intersection(ts) for ss in s for ts in t] ....: temp = filter(lambda x: x != Set([]), temp) ....: return s.__class__(s.parent(), temp)
Let us check that this gives the same as
__mul__
on set partitions of \(\{1, 2, 3, 4\}\):sage: all( all( mul2(s, t) == s * t for s in SetPartitions(4) ) ....: for t in SetPartitions(4) ) True
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is_atomic
()¶ Return if
self
is an atomic set partition.A (standard) set partition \(A\) can be split if there exist \(j < i\) such that \(\max(A_j) < \min(A_i)\) where \(A\) is ordered by minimal elements. This means we can write \(A = B | C\) for some nonempty set partitions \(B\) and \(C\). We call a set partition atomic if it cannot be split and is nonempty. Here, the pipe symbol \(|\) is as defined in method
pipe()
.EXAMPLES:
sage: SetPartition([[1,3], [2]]).is_atomic() True sage: SetPartition([[1,3], [2], [4]]).is_atomic() False sage: SetPartition([[1], [2,4], [3]]).is_atomic() False sage: SetPartition([[1,2,3,4]]).is_atomic() True sage: SetPartition([[1, 4], [2], [3]]).is_atomic() True sage: SetPartition([]).is_atomic() False
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is_noncrossing
()¶ Check if
self
is noncrossing.EXAMPLES:
sage: x = SetPartition([[1,2],[3,4]]) sage: x.is_noncrossing() True sage: x = SetPartition([[1,3],[2,4]]) sage: x.is_noncrossing() False
AUTHOR: Florent Hivert
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ordered_set_partition_action
(s)¶ Return the action of an ordered set partition
s
onself
.Let \(A = \{A_1, A_2, \ldots, A_k\}\) be a set partition of some set \(S\) and \(s\) be an ordered set partition (i.e., set composition) of a subset of \([k]\). Let \(A^{\downarrow}\) denote the standardization of \(A\), and \(A_{\{ i_1, i_2, \ldots, i_m \}}\) denote the sub-partition \(\{A_{i_1}, A_{i_2}, \ldots, A_{i_m}\}\) for any subset \(\{i_1, \ldots, i_m\}\) of \(\{1, \ldots, k\}\). We define the set partition \(s(A)\) by
\[s(A) = A_{s_1}^{\downarrow} | A_{s_2}^{\downarrow} | \cdots | A_{s_q}^{\downarrow}.\]where \(s = (s_1, s_2, \ldots, s_q)\). Here, the pipe symbol \(|\) is as defined in method
pipe()
.This is \(s[A]\) in section 2.3 in [LM2011].
INPUT:
s
– an ordered set partition with base set a subset of \(\{1, \ldots, k\}\)
EXAMPLES:
sage: A = SetPartition([[1], [2,4], [3]]) sage: s = OrderedSetPartition([[1,3], [2]]) sage: A.ordered_set_partition_action(s) {{1}, {2}, {3, 4}} sage: s = OrderedSetPartition([[2,3], [1]]) sage: A.ordered_set_partition_action(s) {{1, 3}, {2}, {4}}
We create Figure 1 in [LM2011] (we note that there is a typo in the lower-left corner of the table in the published version of the paper, whereas the arXiv version gives the correct partition):
sage: A = SetPartition([[1,3], [2,9], [4,5,8], [7]]) sage: B = SetPartition([[1,3], [2,8], [4,5,6], [7]]) sage: C = SetPartition([[1,5], [2,8], [3,4,6], [7]]) sage: s = OrderedSetPartition([[1,3], [2]]) sage: t = OrderedSetPartition([[2], [3,4]]) sage: u = OrderedSetPartition([[1], [2,3,4]]) sage: A.ordered_set_partition_action(s) {{1, 2}, {3, 4, 5}, {6, 7}} sage: A.ordered_set_partition_action(t) {{1, 2}, {3, 4, 6}, {5}} sage: A.ordered_set_partition_action(u) {{1, 2}, {3, 8}, {4, 5, 7}, {6}} sage: B.ordered_set_partition_action(s) {{1, 2}, {3, 4, 5}, {6, 7}} sage: B.ordered_set_partition_action(t) {{1, 2}, {3, 4, 5}, {6}} sage: B.ordered_set_partition_action(u) {{1, 2}, {3, 8}, {4, 5, 6}, {7}} sage: C.ordered_set_partition_action(s) {{1, 4}, {2, 3, 5}, {6, 7}} sage: C.ordered_set_partition_action(t) {{1, 2}, {3, 4, 5}, {6}} sage: C.ordered_set_partition_action(u) {{1, 2}, {3, 8}, {4, 5, 6}, {7}}
REFERENCES:
[LM2011] (1, 2) A. Lauve, M. Mastnak. The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables. Advances in Applied Mathematics. 47 (2011). 536-544. Arxiv 1006.0367v3 doi:10.1016/j.aam.2011.01.002.
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pipe
(other)¶ Return the pipe of the set partitions
self
andother
.The pipe of two set partitions is defined as follows:
For any integer \(k\) and any subset \(I\) of \(\ZZ\), let \(I + k\) denote the subset of \(\ZZ\) obtained by adding \(k\) to every element of \(k\).
If \(B\) and \(C\) are set partitions of \([n]\) and \([m]\), respectively, then the pipe of \(B\) and \(C\) is defined as the set partition
\[\{ B_1, B_2, \ldots, B_b, C_1 + n, C_2 + n, \ldots, C_c + n \}\]of \([n+m]\), where \(B = \{ B_1, B_2, \ldots, B_b \}\) and \(C = \{ C_1, C_2, \ldots, C_c \}\). This pipe is denoted by \(B | C\).
EXAMPLES:
sage: SetPartition([[1,3],[2,4]]).pipe(SetPartition([[1,3],[2]])) {{1, 3}, {2, 4}, {5, 7}, {6}} sage: SetPartition([]).pipe(SetPartition([[1,2],[3,5],[4]])) {{1, 2}, {3, 5}, {4}} sage: SetPartition([[1,2],[3,5],[4]]).pipe(SetPartition([])) {{1, 2}, {3, 5}, {4}} sage: SetPartition([[1,2],[3]]).pipe(SetPartition([[1]])) {{1, 2}, {3}, {4}}
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refinements
()¶ Return a list of refinements of
self
.EXAMPLES:
sage: SetPartition([[1,3],[2,4]]).refinements() [{{1, 3}, {2, 4}}, {{1, 3}, {2}, {4}}, {{1}, {2, 4}, {3}}, {{1}, {2}, {3}, {4}}] sage: SetPartition([[1],[2,4],[3]]).refinements() [{{1}, {2, 4}, {3}}, {{1}, {2}, {3}, {4}}] sage: SetPartition([]).refinements() [{}]
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restriction
(I)¶ Return the restriction of
self
to a subsetI
(which is given as a set or list or any other iterable).EXAMPLES:
sage: A = SetPartition([[1], [2,3]]) sage: A.restriction([1,2]) {{1}, {2}} sage: A.restriction([2,3]) {{2, 3}} sage: A.restriction([]) {} sage: A.restriction([4]) {}
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shape
()¶ Return the integer partition whose parts are the sizes of the sets in
self
.EXAMPLES:
sage: S = SetPartitions(5) sage: x = S([[1,2], [3,5,4]]) sage: x.shape() [3, 2] sage: y = S([[2], [3,1], [5,4]]) sage: y.shape() [2, 2, 1]
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shape_partition
()¶ Return the integer partition whose parts are the sizes of the sets in
self
.EXAMPLES:
sage: S = SetPartitions(5) sage: x = S([[1,2], [3,5,4]]) sage: x.shape() [3, 2] sage: y = S([[2], [3,1], [5,4]]) sage: y.shape() [2, 2, 1]
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size
()¶ Return the cardinality of the base set of
self
, which is the sum of the sizes of the parts ofself
.This is also known as the size (sometimes the weight) of a set partition.
EXAMPLES:
sage: SetPartition([[1], [2,3], [4]]).base_set_cardinality() 4 sage: SetPartition([[1,2,3,4]]).base_set_cardinality() 4
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standard_form
()¶ Return
self
as a list of lists.When the ground set is totally ordered, the elements of each block are listed in increasing order.
This is not related to standard set partitions (which simply means set partitions of \([n] = \{ 1, 2, \ldots , n \}\) for some integer \(n\)) or standardization (
standardization()
).EXAMPLES:
sage: [x.standard_form() for x in SetPartitions(4, [2,2])] [[[1, 2], [3, 4]], [[1, 3], [2, 4]], [[1, 4], [2, 3]]]
TESTS:
sage: SetPartition([(1, 9, 8), (2, 3, 4, 5, 6, 7)]).standard_form() [[1, 8, 9], [2, 3, 4, 5, 6, 7]]
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standardization
()¶ Return the standardization of
self
.Given a set partition \(A = \{A_1, \ldots, A_n\}\) of an ordered set \(S\), the standardization of \(A\) is the set partition of \(\{1, 2, \ldots, |S|\}\) obtained by replacing the elements of the parts of \(A\) by the integers \(1, 2, \ldots, |S|\) in such a way that their relative order is preserved (i. e., the smallest element in the whole set partition is replaced by \(1\), the next-smallest by \(2\), and so on).
EXAMPLES:
sage: SetPartition([[4], [1, 3]]).standardization() {{1, 2}, {3}} sage: SetPartition([[4], [6, 3]]).standardization() {{1, 3}, {2}} sage: SetPartition([]).standardization() {}
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strict_coarsenings
()¶ Return all strict coarsenings of
self
.Strict coarsening is the binary relation on set partitions defined as the transitive-and-reflexive closure of the relation \(\prec\) defined as follows: For two set partitions \(A\) and \(B\), we have \(A \prec B\) if there exist parts \(A_i, A_j\) of \(A\) such that \(\max(A_i) < \min(A_j)\) and \(B = A \setminus \{A_i, A_j\} \cup \{ A_i \cup A_j \}\).
EXAMPLES:
sage: A = SetPartition([[1],[2,3],[4]]) sage: A.strict_coarsenings() [{{1}, {2, 3}, {4}}, {{1, 2, 3}, {4}}, {{1, 4}, {2, 3}}, {{1}, {2, 3, 4}}, {{1, 2, 3, 4}}] sage: SetPartition([[1],[2,4],[3]]).strict_coarsenings() [{{1}, {2, 4}, {3}}, {{1, 2, 4}, {3}}, {{1, 3}, {2, 4}}] sage: SetPartition([]).strict_coarsenings() [{}]
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sup
(t)¶ Return the supremum of
self
andt
in the classical set partition lattice.The supremum of two set partitions \(B\) and \(C\) is obtained as the transitive closure of the relation which relates \(i\) to \(j\) if and only if \(i\) and \(j\) are in the same part in at least one of the set partitions \(B\) and \(C\).
See also
__mul__()
EXAMPLES:
sage: S = SetPartitions(4) sage: sp1 = S([[2,3,4], [1]]) sage: sp2 = S([[1,3], [2,4]]) sage: s = S([[1,2,3,4]]) sage: sp1.sup(sp2) == s True
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to_partition
()¶ Return the integer partition whose parts are the sizes of the sets in
self
.EXAMPLES:
sage: S = SetPartitions(5) sage: x = S([[1,2], [3,5,4]]) sage: x.shape() [3, 2] sage: y = S([[2], [3,1], [5,4]]) sage: y.shape() [2, 2, 1]
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to_permutation
()¶ Convert a set partition of \(\{1,...,n\}\) to a permutation by considering the blocks of the partition as cycles.
The cycles are such that the number of excedences is maximised, that is, each cycle is of the form \((a_1,a_2, ...,a_k)\) with \(a_1<a_2<...<a_k\).
EXAMPLES:
sage: s = SetPartition([[1,3],[2,4]]) sage: s.to_permutation() [3, 4, 1, 2]
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class
sage.combinat.set_partition.
SetPartitions
¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
An (unordered) partition of a set \(S\) is a set of pairwise disjoint nonempty subsets with union \(S\), and is represented by a sorted list of such subsets.
SetPartitions(s)
returns the class of all set partitions of the sets
, which can be given as a set or a string; if a string, each character is considered an element.SetPartitions(n)
, wheren
is an integer, returns the class of all set partitions of the set \(\{1, 2, \ldots, n\}\).You may specify a second argument \(k\). If \(k\) is an integer,
SetPartitions
returns the class of set partitions into \(k\) parts; if it is an integer partition,SetPartitions
returns the class of set partitions whose block sizes correspond to that integer partition.The Bell number \(B_n\), named in honor of Eric Temple Bell, is the number of different partitions of a set with \(n\) elements.
EXAMPLES:
sage: S = [1,2,3,4] sage: SetPartitions(S,2) Set partitions of {1, 2, 3, 4} with 2 parts sage: SetPartitions([1,2,3,4], [3,1]).list() [{{1}, {2, 3, 4}}, {{1, 3, 4}, {2}}, {{1, 2, 4}, {3}}, {{1, 2, 3}, {4}}] sage: SetPartitions(7, [3,3,1]).cardinality() 70
In strings, repeated letters are not considered distinct as of trac ticket #14140:
sage: SetPartitions('abcde').cardinality() 52 sage: SetPartitions('aabcd').cardinality() 15
REFERENCES:
-
Element
¶ alias of
SetPartition
-
is_less_than
(s, t)¶ Check if \(s < t\) in the refinement ordering on set partitions.
This means that \(s\) is a refinement of \(t\) and satisfies \(s \neq t\).
A set partition \(s\) is said to be a refinement of a set partition \(t\) of the same set if and only if each part of \(s\) is a subset of a part of \(t\).
EXAMPLES:
sage: S = SetPartitions(4) sage: s = S([[1,3],[2,4]]) sage: t = S([[1],[2],[3],[4]]) sage: S.is_less_than(t, s) True sage: S.is_less_than(s, t) False sage: S.is_less_than(s, s) False
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is_strict_refinement
(s, t)¶ Return
True
ifs
is a strict refinement oft
and satisfies \(s \neq t\).A set partition \(s\) is said to be a strict refinement of a set partition \(t\) of the same set if and only if one can obtain \(t\) from \(s\) by repeatedly combining pairs of parts whose convex hulls don’t intersect (i. e., whenever we are combining two parts, the maximum of each of them should be smaller than the minimum of the other).
EXAMPLES:
sage: S = SetPartitions(4) sage: s = S([[1],[2],[3],[4]]) sage: t = S([[1,3],[2,4]]) sage: u = S([[1,2,3,4]]) sage: S.is_strict_refinement(s, t) True sage: S.is_strict_refinement(t, u) False sage: A = SetPartition([[1,3],[2,4]]) sage: B = SetPartition([[1,2,3,4]]) sage: S.is_strict_refinement(s, A) True sage: S.is_strict_refinement(t, B) False
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lt
(s, t)¶ Check if \(s < t\) in the refinement ordering on set partitions.
This means that \(s\) is a refinement of \(t\) and satisfies \(s \neq t\).
A set partition \(s\) is said to be a refinement of a set partition \(t\) of the same set if and only if each part of \(s\) is a subset of a part of \(t\).
EXAMPLES:
sage: S = SetPartitions(4) sage: s = S([[1,3],[2,4]]) sage: t = S([[1],[2],[3],[4]]) sage: S.is_less_than(t, s) True sage: S.is_less_than(s, t) False sage: S.is_less_than(s, s) False
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-
class
sage.combinat.set_partition.
SetPartitions_all
¶ Bases:
sage.combinat.set_partition.SetPartitions
All set partitions.
-
class
sage.combinat.set_partition.
SetPartitions_set
(s)¶ Bases:
sage.combinat.set_partition.SetPartitions
Set partitions of a fixed set \(S\).
-
base_set
()¶ Return the base set of
self
.EXAMPLES:
sage: SetPartitions(3).base_set() {1, 2, 3}
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base_set_cardinality
()¶ Return the cardinality of the base set of
self
.EXAMPLES:
sage: SetPartitions(3).base_set_cardinality() 3
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cardinality
()¶ Return the number of set partitions of the set \(S\).
The cardinality is given by the \(n\)-th Bell number where \(n\) is the number of elements in the set \(S\).
EXAMPLES:
sage: SetPartitions([1,2,3,4]).cardinality() 15 sage: SetPartitions(3).cardinality() 5 sage: SetPartitions(3,2).cardinality() 3 sage: SetPartitions([]).cardinality() 1
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-
class
sage.combinat.set_partition.
SetPartitions_setn
(s, n)¶ Bases:
sage.combinat.set_partition.SetPartitions_set
TESTS:
sage: S = SetPartitions(5, 3) sage: TestSuite(S).run()
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cardinality
()¶ The Stirling number of the second kind is the number of partitions of a set of size \(n\) into \(k\) blocks.
EXAMPLES:
sage: SetPartitions(5, 3).cardinality() 25 sage: stirling_number2(5,3) 25
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-
class
sage.combinat.set_partition.
SetPartitions_setparts
(s, parts)¶ Bases:
sage.combinat.set_partition.SetPartitions_set
Class of all set partitions with fixed partition sizes corresponding to an integer partition \(\lambda\).
-
cardinality
()¶ Return the cardinality of
self
.This algorithm counts for each block of the partition the number of ways to fill it using values from the set. Then, for each distinct value \(v\) of block size, we divide the result by the number of ways to arrange the blocks of size \(v\) in the set partition.
For example, if we want to count the number of set partitions of size 13 having [3,3,3,2,2] as underlying partition we compute the number of ways to fill each block of the partition, which is \(\binom{13}{3} \binom{10}{3} \binom{7}{3} \binom{4}{2}\binom{2}{2}\) and as we have three blocks of size \(3\) and two blocks of size \(2\), we divide the result by \(3!2!\) which gives us \(600600\).
EXAMPLES:
sage: SetPartitions(3, [2,1]).cardinality() 3 sage: SetPartitions(13, Partition([3,3,3,2,2])).cardinality() 600600
TESTS:
sage: all((len(SetPartitions(size, part)) == SetPartitions(size, part).cardinality() for size in range(8) for part in Partitions(size))) True sage: sum((SetPartitions(13, p).cardinality() for p in Partitions(13))) == SetPartitions(13).cardinality() True
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-
sage.combinat.set_partition.
cyclic_permutations_of_set_partition
(set_part)¶ Returns all combinations of cyclic permutations of each cell of the set partition.
AUTHORS:
- Robert L. Miller
EXAMPLES:
sage: from sage.combinat.set_partition import cyclic_permutations_of_set_partition sage: cyclic_permutations_of_set_partition([[1,2,3,4],[5,6,7]]) [[[1, 2, 3, 4], [5, 6, 7]], [[1, 2, 4, 3], [5, 6, 7]], [[1, 3, 2, 4], [5, 6, 7]], [[1, 3, 4, 2], [5, 6, 7]], [[1, 4, 2, 3], [5, 6, 7]], [[1, 4, 3, 2], [5, 6, 7]], [[1, 2, 3, 4], [5, 7, 6]], [[1, 2, 4, 3], [5, 7, 6]], [[1, 3, 2, 4], [5, 7, 6]], [[1, 3, 4, 2], [5, 7, 6]], [[1, 4, 2, 3], [5, 7, 6]], [[1, 4, 3, 2], [5, 7, 6]]]
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sage.combinat.set_partition.
cyclic_permutations_of_set_partition_iterator
(set_part)¶ Iterates over all combinations of cyclic permutations of each cell of the set partition.
AUTHORS:
- Robert L. Miller
EXAMPLES:
sage: from sage.combinat.set_partition import cyclic_permutations_of_set_partition_iterator sage: list(cyclic_permutations_of_set_partition_iterator([[1,2,3,4],[5,6,7]])) [[[1, 2, 3, 4], [5, 6, 7]], [[1, 2, 4, 3], [5, 6, 7]], [[1, 3, 2, 4], [5, 6, 7]], [[1, 3, 4, 2], [5, 6, 7]], [[1, 4, 2, 3], [5, 6, 7]], [[1, 4, 3, 2], [5, 6, 7]], [[1, 2, 3, 4], [5, 7, 6]], [[1, 2, 4, 3], [5, 7, 6]], [[1, 3, 2, 4], [5, 7, 6]], [[1, 3, 4, 2], [5, 7, 6]], [[1, 4, 2, 3], [5, 7, 6]], [[1, 4, 3, 2], [5, 7, 6]]]