Vector Partitions

AUTHORS:

• Amritanshu Prasad (2013): Initial version

• Shriya M (2022): Added new parameters such as distinct, parts and is_repeatable

sage.combinat.vector_partition.IntegerVectorsIterator(vect, min=None)

Return an iterator over the list of integer vectors which are componentwise less than or equal to vect, and lexicographically greater than or equal to min.

INPUT:

• vect – list of nonnegative integers

• min – list of nonnegative integers dominated elementwise by vect

OUTPUT:

A list in lexicographic order of all integer vectors (as lists) which are dominated elementwise by vect and are greater than or equal to min in lexicographic order.

EXAMPLES:

Sage

sage: from sage.combinat.vector_partition import IntegerVectorsIterator
sage: list(IntegerVectorsIterator([1, 1]))
[[0, 0], [0, 1], [1, 0], [1, 1]]

sage: list(IntegerVectorsIterator([1, 1], min = [1, 0]))
[[1, 0], [1, 1]]

Python

>>> from sage.all import *
>>> from sage.combinat.vector_partition import IntegerVectorsIterator
>>> list(IntegerVectorsIterator([Integer(1), Integer(1)]))
[[0, 0], [0, 1], [1, 0], [1, 1]]

>>> list(IntegerVectorsIterator([Integer(1), Integer(1)], min = [Integer(1), Integer(0)]))
[[1, 0], [1, 1]]

class sage.combinat.vector_partition.VectorPartition(parent, vecpar)

Bases: CombinatorialElement

A vector partition is a multiset of integer vectors.

partition_at_vertex(i)

Return the partition obtained by sorting the i-th elements of the vectors in the vector partition.

EXAMPLES:

Sage

sage: V = VectorPartition([[1, 2, 1], [2, 4, 1]])
sage: V.partition_at_vertex(1)
[4, 2]

Python

>>> from sage.all import *
>>> V = VectorPartition([[Integer(1), Integer(2), Integer(1)], [Integer(2), Integer(4), Integer(1)]])
>>> V.partition_at_vertex(Integer(1))
[4, 2]

sum()

Return the sum vector as a list.

EXAMPLES:

Sage

sage: V = VectorPartition([[3, 2, 1], [2, 2, 1]])
sage: V.sum()
[5, 4, 2]

Python

>>> from sage.all import *
>>> V = VectorPartition([[Integer(3), Integer(2), Integer(1)], [Integer(2), Integer(2), Integer(1)]])
>>> V.sum()
[5, 4, 2]

class sage.combinat.vector_partition.VectorPartitions(vec, min=None, parts=None, distinct=False, is_repeatable=None)

Bases: UniqueRepresentation, Parent

Class of all vector partitions of vec with all parts greater than or equal to min in lexicographic order, with parts from parts.

A vector partition of vec is a list of vectors with nonnegative integer entries whose sum is vec.

INPUT:

• vec – integer vector

• min – integer vector dominated elementwise by vec

• parts – finite list of possible parts

• distinct – boolean, set to True if only vector partitions with distinct parts are enumerated

• is_repeatable – boolean function on parts which gives True in parts that can be repeated

EXAMPLES:

If min is not specified, then the class of all vector partitions of vec is created:

Sage

sage: VP = VectorPartitions([2, 2])
sage: for vecpar in VP:
....:     print(vecpar)
[[0, 1], [0, 1], [1, 0], [1, 0]]
[[0, 1], [0, 1], [2, 0]]
[[0, 1], [1, 0], [1, 1]]
[[0, 1], [2, 1]]
[[0, 2], [1, 0], [1, 0]]
[[0, 2], [2, 0]]
[[1, 0], [1, 2]]
[[1, 1], [1, 1]]
[[2, 2]]

Python

>>> from sage.all import *
>>> VP = VectorPartitions([Integer(2), Integer(2)])
>>> for vecpar in VP:
...     print(vecpar)
[[0, 1], [0, 1], [1, 0], [1, 0]]
[[0, 1], [0, 1], [2, 0]]
[[0, 1], [1, 0], [1, 1]]
[[0, 1], [2, 1]]
[[0, 2], [1, 0], [1, 0]]
[[0, 2], [2, 0]]
[[1, 0], [1, 2]]
[[1, 1], [1, 1]]
[[2, 2]]

If distinct is set to be True, then distinct part partitions are created:

Sage

sage: VP = VectorPartitions([2,2], distinct = True)
sage: list(VP)
[[[0, 1], [1, 0], [1, 1]],
 [[0, 1], [2, 1]],
 [[0, 2], [2, 0]],
 [[1, 0], [1, 2]],
 [[2, 2]]]

Python

>>> from sage.all import *
>>> VP = VectorPartitions([Integer(2),Integer(2)], distinct = True)
>>> list(VP)
[[[0, 1], [1, 0], [1, 1]],
 [[0, 1], [2, 1]],
 [[0, 2], [2, 0]],
 [[1, 0], [1, 2]],
 [[2, 2]]]

If min is specified, then the class consists of only those vector partitions whose parts are all greater than or equal to min in lexicographic order:

Sage

sage: VP = VectorPartitions([2, 2], min = [1, 0])
sage: for vecpar in VP:
....:     print(vecpar)
[[1, 0], [1, 2]]
[[1, 1], [1, 1]]
[[2, 2]]
sage: VP = VectorPartitions([2, 2], min = [1, 0], distinct = True)
sage: for vecpar in VP:
....:     print(vecpar)
[[1, 0], [1, 2]]
[[2, 2]]

Python

>>> from sage.all import *
>>> VP = VectorPartitions([Integer(2), Integer(2)], min = [Integer(1), Integer(0)])
>>> for vecpar in VP:
...     print(vecpar)
[[1, 0], [1, 2]]
[[1, 1], [1, 1]]
[[2, 2]]
>>> VP = VectorPartitions([Integer(2), Integer(2)], min = [Integer(1), Integer(0)], distinct = True)
>>> for vecpar in VP:
...     print(vecpar)
[[1, 0], [1, 2]]
[[2, 2]]

If parts is specified, then the class consists only of those vector partitions whose parts are from parts:

Sage

sage: Vec_Par = VectorPartitions([2,2], parts=[[0,1],[1,0],[1,1]])
sage: list(Vec_Par)
[[[0, 1], [0, 1], [1, 0], [1, 0]], [[0, 1], [1, 0], [1, 1]], [[1, 1], [1, 1]]]

Python

>>> from sage.all import *
>>> Vec_Par = VectorPartitions([Integer(2),Integer(2)], parts=[[Integer(0),Integer(1)],[Integer(1),Integer(0)],[Integer(1),Integer(1)]])
>>> list(Vec_Par)
[[[0, 1], [0, 1], [1, 0], [1, 0]], [[0, 1], [1, 0], [1, 1]], [[1, 1], [1, 1]]]

If is_repeatable is specified, then the parts which satisfy the boolean function is_repeatable are allowed to be repeated:

Sage

sage: Vector_Partitions = VectorPartitions([2,2], parts=[[0,1],[1,0],[1,1]], is_repeatable=lambda vec: sum(vec)%2!=0)
sage: list(Vector_Partitions)
[[[0, 1], [0, 1], [1, 0], [1, 0]], [[0, 1], [1, 0], [1, 1]]]

Python

>>> from sage.all import *
>>> Vector_Partitions = VectorPartitions([Integer(2),Integer(2)], parts=[[Integer(0),Integer(1)],[Integer(1),Integer(0)],[Integer(1),Integer(1)]], is_repeatable=lambda vec: sum(vec)%Integer(2)!=Integer(0))
>>> list(Vector_Partitions)
[[[0, 1], [0, 1], [1, 0], [1, 0]], [[0, 1], [1, 0], [1, 1]]]

Element

alias of VectorPartition

sage.combinat.vector_partition.find_min(vect)

Return a string of 0’s with one 1 at the location where the list vect has its last entry which is not equal to 0.

INPUT:

• vec – list of integers

OUTPUT:

A list of the same length with 0’s everywhere, except for a 1 at the last position where vec has an entry not equal to 0.

EXAMPLES:

Sage

sage: from sage.combinat.vector_partition import find_min
sage: find_min([2, 1])
[0, 1]
sage: find_min([2, 1, 0])
[0, 1, 0]

Python

>>> from sage.all import *
>>> from sage.combinat.vector_partition import find_min
>>> find_min([Integer(2), Integer(1)])
[0, 1]
>>> find_min([Integer(2), Integer(1), Integer(0)])
[0, 1, 0]