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
. - Martin Rubey (2017-10-10): Cleanup, add crossings and nestings, add random generation.
This module defines a class for immutable partitioning of a set. For
mutable version see DisjointSet()
.
-
sage.combinat.set_partition.
AbstractSetPartition
¶ Methods of set partitions which are independent of the base set
-
sage.combinat.set_partition.
SetPartition
¶ 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
-
sage.combinat.set_partition.
SetPartitions
¶ 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:
-
sage.combinat.set_partition.
SetPartitions_all
¶ All set partitions.
-
sage.combinat.set_partition.
SetPartitions_set
¶ Set partitions of a fixed set \(S\).
-
sage.combinat.set_partition.
SetPartitions_setn
¶ Set partitions with a given number of blocks.
-
sage.combinat.set_partition.
SetPartitions_setparts
¶ Set partitions with fixed partition sizes corresponding to an integer partition \(\lambda\).
-
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]]]
-
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]]]