Partitions¶

class
diofant.combinatorics.partitions.
Partition
[source]¶ This class represents an abstract partition.
A partition is a set of disjoint sets whose union equals a given set.

RGS
¶ Returns the “restricted growth string” of the partition.
The RGS is returned as a list of indices, L, where L[i] indicates the block in which element i appears. For example, in a partition of 3 elements (a, b, c) into 2 blocks ([c], [a, b]) the RGS is [1, 1, 0]: “a” is in block 1, “b” is in block 1 and “c” is in block 0.
Examples
>>> a = Partition([1, 2], [3], [4, 5]) >>> a.members (1, 2, 3, 4, 5) >>> a.RGS (0, 0, 1, 2, 2) >>> a + 1 {{3}, {4}, {5}, {1, 2}} >>> _.RGS (0, 0, 1, 2, 3)

classmethod
from_rgs
(rgs, elements)[source]¶ Creates a set partition from a restricted growth string.
The indices given in rgs are assumed to be the index of the element as given in elements as provided (the elements are not sorted by this routine). Block numbering starts from 0. If any block was not referenced in
rgs
an error will be raised.Examples
>>> Partition.from_rgs([0, 1, 2, 0, 1], list('abcde')) {{c}, {a, d}, {b, e}} >>> Partition.from_rgs([0, 1, 2, 0, 1], list('cbead')) {{e}, {a, c}, {b, d}} >>> a = Partition([1, 4], [2], [3, 5]) >>> Partition.from_rgs(a.RGS, a.members) {{2}, {1, 4}, {3, 5}}

partition
¶ Return partition as a sorted list of lists.
Examples
>>> Partition([1], [2, 3]).partition [[1], [2, 3]]

rank
¶ Gets the rank of a partition.
Examples
>>> a = Partition([1, 2], [3], [4, 5]) >>> a.rank 13

sort_key
(order=None)[source]¶ Return a canonical key that can be used for sorting.
Ordering is based on the size and sorted elements of the partition and ties are broken with the rank.
Examples
>>> a = Partition([1, 2]) >>> b = Partition([3, 4]) >>> c = Partition([1, x]) >>> d = Partition(list(range(4))) >>> l = [d, b, a + 1, a, c] >>> l.sort(key=default_sort_key); l [{{1, 2}}, {{1}, {2}}, {{1, x}}, {{3, 4}}, {{0, 1, 2, 3}}]


class
diofant.combinatorics.partitions.
IntegerPartition
[source]¶ This class represents an integer partition.
In number theory and combinatorics, a partition of a positive integer,
n
, also called an integer partition, is a way of writingn
as a list of positive integers that sum to n. Two partitions that differ only in the order of summands are considered to be the same partition; if order matters then the partitions are referred to as compositions. For example, 4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1]; the compositions [1, 2, 1] and [1, 1, 2] are the same as partition [2, 1, 1].References

as_dict
()[source]¶ Return the partition as a dictionary whose keys are the partition integers and the values are the multiplicity of that integer.
Examples
>>> IntegerPartition([1]*3 + [2] + [3]*4).as_dict() {1: 3, 2: 1, 3: 4}

as_ferrers
(char='#')[source]¶ Prints the ferrer diagram of a partition.
Examples
>>> print(IntegerPartition([1, 1, 5]).as_ferrers()) ##### # #

conjugate
¶ Computes the conjugate partition of itself.
Examples
>>> a = IntegerPartition([6, 3, 3, 2, 1]) >>> a.conjugate [5, 4, 3, 1, 1, 1]


diofant.combinatorics.partitions.
random_integer_partition
(n, seed=None)[source]¶ Generates a random integer partition summing to
n
as a list of reversesorted integers.Examples
For the following, a seed is given so a known value can be shown; in practice, the seed would not be given.
>>> random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1]) [85, 12, 2, 1] >>> random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1]) [5, 3, 1, 1] >>> random_integer_partition(1) [1]

diofant.combinatorics.partitions.
RGS_generalized
(m)[source]¶ Computes the m + 1 generalized unrestricted growth strings and returns them as rows in matrix.
Examples
>>> RGS_generalized(6) Matrix([ [ 1, 1, 1, 1, 1, 1, 1], [ 1, 2, 3, 4, 5, 6, 0], [ 2, 5, 10, 17, 26, 0, 0], [ 5, 15, 37, 77, 0, 0, 0], [ 15, 52, 151, 0, 0, 0, 0], [ 52, 203, 0, 0, 0, 0, 0], [203, 0, 0, 0, 0, 0, 0]])

diofant.combinatorics.partitions.
RGS_enum
(m)[source]¶ RGS_enum computes the total number of restricted growth strings possible for a superset of size m.
Examples
>>> RGS_enum(4) 15 >>> RGS_enum(5) 52 >>> RGS_enum(6) 203
We can check that the enumeration is correct by actually generating the partitions. Here, the 15 partitions of 4 items are generated:
>>> a = Partition(list(range(4))) >>> s = set() >>> for i in range(20): ... s.add(a) ... a += 1 ... >>> assert len(s) == 15