(Non-negative) Integer vectors¶
AUTHORS:
- Mike Hansen (2007) - original module
- Nathann Cohen, David Joyner (2009-2010) - Gale-Ryser stuff
- Nathann Cohen, David Joyner (2011) - Gale-Ryser bugfix
- Travis Scrimshaw (2012-05-12) - Updated doc-strings to tell the user of that the class’s name is a misnomer (that they only contains non-negative entries).
- Federico Poloni (2013) - specialized rank()
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sage.combinat.integer_vector.
IntegerVectors
(n=None, k=None, **kwargs)¶ Returns the combinatorial class of (non-negative) integer vectors.
INPUT:
- \(n\) – if set to an integer, returns the combinatorial class of integer
vectors whose sum is \(n\). If set to
None
(default), no such constraint is defined. k
– the length of the vectors. Set toNone
(default) if you do not want such a constraint.
All other arguments given to this function are forwarded to the instance of
IntegerVectors_all
IntegerVectors_nconstraints
IntegerVectors_nk
orIntegerVectors_nkconstraints
that it returns.NOTE - These integer vectors are non-negative.
EXAMPLES: If n is not specified, it returns the class of all integer vectors.
sage: IntegerVectors() Integer vectors sage: [] in IntegerVectors() True sage: [1,2,1] in IntegerVectors() True sage: [1, 0, 0] in IntegerVectors() True
Entries are non-negative.
sage: [-1, 2] in IntegerVectors() False
If n is specified, then it returns the class of all integer vectors which sum to n.
sage: IV3 = IntegerVectors(3); IV3 Integer vectors that sum to 3
Note that trailing zeros are ignored so that [3, 0] does not show up in the following list (since [3] does)
sage: IntegerVectors(3, max_length=2).list() [[3], [2, 1], [1, 2], [0, 3]]
If n and k are both specified, then it returns the class of integer vectors that sum to n and are of length k.
sage: IV53 = IntegerVectors(5,3); IV53 Integer vectors of length 3 that sum to 5 sage: IV53.cardinality() 21 sage: IV53.first() [5, 0, 0] sage: IV53.last() [0, 0, 5] sage: IV53.random_element() [4, 0, 1]
Further examples:
sage: IntegerVectors(-1, 0, min_part = 1).list() [] sage: IntegerVectors(-1, 2, min_part = 1).list() [] sage: IntegerVectors(0, 0, min_part=1).list() [[]] sage: IntegerVectors(3, 0, min_part=1).list() [] sage: IntegerVectors(0, 1, min_part=1).list() [] sage: IntegerVectors(2, 2, min_part=1).list() [[1, 1]] sage: IntegerVectors(2, 3, min_part=1).list() [] sage: IntegerVectors(4, 2, min_part=1).list() [[3, 1], [2, 2], [1, 3]]
sage: IntegerVectors(0, 3, outer=[0,0,0]).list() [[0, 0, 0]] sage: IntegerVectors(1, 3, outer=[0,0,0]).list() [] sage: IntegerVectors(2, 3, outer=[0,2,0]).list() [[0, 2, 0]] sage: IntegerVectors(2, 3, outer=[1,2,1]).list() [[1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1]] sage: IntegerVectors(2, 3, outer=[1,1,1]).list() [[1, 1, 0], [1, 0, 1], [0, 1, 1]] sage: IntegerVectors(2, 5, outer=[1,1,1,1,1]).list() [[1, 1, 0, 0, 0], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0], [1, 0, 0, 0, 1], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]]
sage: iv = [ IntegerVectors(n,k) for n in range(-2, 7) for k in range(7) ] sage: all(map(lambda x: x.cardinality() == len(x.list()), iv)) True sage: essai = [[1,1,1], [2,5,6], [6,5,2]] sage: iv = [ IntegerVectors(x[0], x[1], max_part = x[2]-1) for x in essai ] sage: all(map(lambda x: x.cardinality() == len(x.list()), iv)) True
TESTS:
sage: IntegerVectors(None, length=3) Traceback (most recent call last): ... TypeError: __init__() got an unexpected keyword argument 'length' sage: IntegerVectors(None, 4) Traceback (most recent call last): ... NotImplementedError: k must be None when n is None
- \(n\) – if set to an integer, returns the combinatorial class of integer
vectors whose sum is \(n\). If set to
-
class
sage.combinat.integer_vector.
IntegerVectors_all
(category=None)¶ Bases:
sage.combinat.combinat.CombinatorialClass
TESTS:
sage: C = sage.combinat.combinat.CombinatorialClass() sage: C.category() Category of enumerated sets sage: C.__class__ <class 'sage.combinat.combinat.CombinatorialClass_with_category'> sage: isinstance(C, Parent) True sage: C = sage.combinat.combinat.CombinatorialClass(category = FiniteEnumeratedSets()) sage: C.category() Category of finite enumerated sets
-
cardinality
()¶ EXAMPLES:
sage: IntegerVectors().cardinality() +Infinity
-
list
()¶ EXAMPLES:
sage: IntegerVectors().list() Traceback (most recent call last): ... NotImplementedError: infinite list
-
-
class
sage.combinat.integer_vector.
IntegerVectors_nconstraints
(n, constraints)¶ Bases:
sage.combinat.integer_vector.IntegerVectors_nkconstraints
TESTS:
sage: IV = IntegerVectors(3, max_length=2) sage: IV == loads(dumps(IV)) True sage: IntegerVectors(3, max_length=2).cardinality() 4 sage: IntegerVectors(3).cardinality() +Infinity sage: IntegerVectors(3, max_length=2).list() [[3], [2, 1], [1, 2], [0, 3]] sage: IntegerVectors(3).list() Traceback (most recent call last): ... NotImplementedError: infinite list
-
class
sage.combinat.integer_vector.
IntegerVectors_nk
(n, k)¶ Bases:
sage.combinat.combinat.CombinatorialClass
TESTS:
sage: IV = IntegerVectors(2,3) sage: IV == loads(dumps(IV)) True
AUTHORS:
- Martin Albrecht
- Mike Hansen
-
list
()¶ EXAMPLE:
sage: IV = IntegerVectors(2,3) sage: IV.list() [[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]] sage: IntegerVectors(3, 0).list() [] sage: IntegerVectors(3, 1).list() [[3]] sage: IntegerVectors(0, 1).list() [[0]] sage: IntegerVectors(0, 2).list() [[0, 0]] sage: IntegerVectors(2, 2).list() [[2, 0], [1, 1], [0, 2]]
-
rank
(x)¶ Returns the position of a given element.
INPUT:
x
- a list withsum(x) == n
andlen(x) == k
TESTS:
sage: IV = IntegerVectors(4,5) sage: range(IV.cardinality()) == [IV.rank(x) for x in IV] True
-
class
sage.combinat.integer_vector.
IntegerVectors_nkconstraints
(n, k, constraints, category=None)¶ Bases:
sage.combinat.integer_lists.invlex.IntegerListsLex
EXAMPLES:
sage: IV = IntegerVectors(2,3,min_slope=0) sage: IV == loads(dumps(IV)) True sage: v = IntegerVectors(2,3,min_slope=0).first(); v [0, 1, 1] sage: type(v) <type 'list'>
TESTS:
sage: IV.min_length 3 sage: IV.max_length 3 sage: floor = IV.floor sage: [floor(i) for i in range(1,10)] [0, 0, 0, 0, 0, 0, 0, 0, 0] sage: ceiling = IV.ceiling sage: [ceiling(i) for i in range(1,5)] [inf, inf, inf, inf] sage: IV.min_slope 0 sage: IV.max_slope inf sage: IV = IntegerVectors(3, 10, inner=[4,1,3], min_part=2) sage: floor = IV.floor sage: floor(0), floor(1), floor(2) (4, 2, 3) sage: IV = IntegerVectors(3, 10, outer=[4,1,3], max_part=3) sage: ceiling = IV.ceiling sage: ceiling(0), ceiling(1), ceiling(2) (3, 1, 3)
-
cardinality
()¶ EXAMPLES:
sage: IntegerVectors(3,3, min_part=1).cardinality() 1 sage: IntegerVectors(5,3, min_part=1).cardinality() 6 sage: IntegerVectors(13, 4, min_part=2, max_part=4).cardinality() 16
-
next
(x)¶ EXAMPLES:
sage: a = IntegerVectors(2,3,min_slope=0).first() sage: IntegerVectors(2,3,min_slope=0).next(a) [0, 0, 2]
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-
class
sage.combinat.integer_vector.
IntegerVectors_nnondescents
(n, comp)¶ Bases:
sage.combinat.combinat.CombinatorialClass
The combinatorial class of integer vectors v graded by two parameters:
- n: the sum of the parts of v
- comp: the non descents composition of v
In other words: the length of v equals c[1]+...+c[k], and v is decreasing in the consecutive blocs of length c[1], ..., c[k]
Those are the integer vectors of sum n which are lexicographically maximal (for the natural left->right reading) in their orbit by the young subgroup S_c_1 x x S_c_k. In particular, they form a set of orbit representative of integer vectors w.r.t. this young subgroup.
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sage.combinat.integer_vector.
constant_func
(i)¶ Returns the constant function i.
EXAMPLES:
sage: f = sage.combinat.integer_vector.constant_func(3) sage: f(-1) 3 sage: f('asf') 3
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sage.combinat.integer_vector.
gale_ryser_theorem
(p1, p2, algorithm='gale')¶ Returns the binary matrix given by the Gale-Ryser theorem.
The Gale Ryser theorem asserts that if \(p_1,p_2\) are two partitions of \(n\) of respective lengths \(k_1,k_2\), then there is a binary \(k_1\times k_2\) matrix \(M\) such that \(p_1\) is the vector of row sums and \(p_2\) is the vector of column sums of \(M\), if and only if the conjugate of \(p_2\) dominates \(p_1\).
INPUT:
p1, p2
– list of integers representing the vectors of row/column sumsalgorithm
– two possible string values :
OUTPUT:
- A binary matrix if it exists,
None
otherwise.
Gale’s Algorithm:
(Gale [Gale57]): A matrix satisfying the constraints of its sums can be defined as the solution of the following Linear Program, which Sage knows how to solve.
\[\begin{split}\forall i&\sum_{j=1}^{k_2} b_{i,j}=p_{1,j}\\ \forall i&\sum_{j=1}^{k_1} b_{j,i}=p_{2,j}\\ &b_{i,j}\mbox{ is a binary variable}\end{split}\]Ryser’s Algorithm:
(Ryser [Ryser63]): The construction of an \(m\times n\) matrix \(A=A_{r,s}\), due to Ryser, is described as follows. The construction works if and only if have \(s\preceq r^*\).
- Construct the \(m\times n\) matrix \(B\) from \(r\) by defining the \(i\)-th row of \(B\) to be the vector whose first \(r_i\) entries are \(1\), and the remainder are 0’s, \(1\leq i\leq m\). This maximal matrix \(B\) with row sum \(r\) and ones left justified has column sum \(r^{*}\).
- Shift the last \(1\) in certain rows of \(B\) to column \(n\) in
order to achieve the sum \(s_n\). Call this \(B\) again.
- The \(1\)‘s in column n are to appear in those rows in which \(A\) has the largest row sums, giving preference to the bottom-most positions in case of ties.
- Note: When this step automatically “fixes” other columns, one must skip ahead to the first column index with a wrong sum in the step below.
- Proceed inductively to construct columns \(n-1\), ..., \(2\), \(1\). Note: when performing the induction on step \(k\), we consider the row sums of the first \(k\) columns.
- Set \(A = B\). Return \(A\).
EXAMPLES:
Computing the matrix for \(p_1=p_2=2+2+1\)
sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: p1 = [2,2,1] sage: p2 = [2,2,1] sage: print(gale_ryser_theorem(p1, p2)) # not tested [1 1 0] [1 0 1] [0 1 0] sage: A = gale_ryser_theorem(p1, p2) sage: rs = [sum(x) for x in A.rows()] sage: cs = [sum(x) for x in A.columns()] sage: p1 == rs; p2 == cs True True
Or for a non-square matrix with \(p_1=3+3+2+1\) and \(p_2=3+2+2+1+1\), using Ryser’s algorithm
sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: p1 = [3,3,1,1] sage: p2 = [3,3,1,1] sage: gale_ryser_theorem(p1, p2, algorithm = "ryser") [1 1 1 0] [1 1 0 1] [1 0 0 0] [0 1 0 0] sage: p1 = [4,2,2] sage: p2 = [3,3,1,1] sage: gale_ryser_theorem(p1, p2, algorithm = "ryser") [1 1 1 1] [1 1 0 0] [1 1 0 0] sage: p1 = [4,2,2,0] sage: p2 = [3,3,1,1,0,0] sage: gale_ryser_theorem(p1, p2, algorithm = "ryser") [1 1 1 1 0 0] [1 1 0 0 0 0] [1 1 0 0 0 0] [0 0 0 0 0 0] sage: p1 = [3,3,2,1] sage: p2 = [3,2,2,1,1] sage: print(gale_ryser_theorem(p1, p2, algorithm="gale")) # not tested [1 1 1 0 0] [1 1 0 0 1] [1 0 1 0 0] [0 0 0 1 0]
With \(0\) in the sequences, and with unordered inputs
sage: from sage.combinat.integer_vector import gale_ryser_theorem sage: gale_ryser_theorem([3,3,0,1,1,0], [3,1,3,1,0], algorithm = "ryser") [1 1 1 0 0] [1 0 1 1 0] [0 0 0 0 0] [1 0 0 0 0] [0 0 1 0 0] [0 0 0 0 0] sage: p1 = [3,1,1,1,1]; p2 = [3,2,2,0] sage: gale_ryser_theorem(p1, p2, algorithm = "ryser") [1 1 1 0] [1 0 0 0] [1 0 0 0] [0 1 0 0] [0 0 1 0]
TESTS:
This test created a random bipartite graph on \(n+m\) vertices. Its adjacency matrix is binary, and it is used to create some “random-looking” sequences which correspond to an existing matrix. The
gale_ryser_theorem
is then called on these sequences, and the output checked for correctness.:sage: def test_algorithm(algorithm, low = 10, high = 50): ....: n,m = randint(low,high), randint(low,high) ....: g = graphs.RandomBipartite(n, m, .3) ....: s1 = sorted(g.degree([(0,i) for i in range(n)]), reverse = True) ....: s2 = sorted(g.degree([(1,i) for i in range(m)]), reverse = True) ....: m = gale_ryser_theorem(s1, s2, algorithm = algorithm) ....: ss1 = sorted(map(lambda x : sum(x) , m.rows()), reverse = True) ....: ss2 = sorted(map(lambda x : sum(x) , m.columns()), reverse = True) ....: if ((ss1 != s1) or (ss2 != s2)): ....: print("Algorithm %s failed with this input:" % algorithm) ....: print(s1, s2) sage: for algorithm in ["gale", "ryser"]: # long time ....: for i in range(50): # long time ....: test_algorithm(algorithm, 3, 10) # long time
Null matrix:
sage: gale_ryser_theorem([0,0,0],[0,0,0,0], algorithm="gale") [0 0 0 0] [0 0 0 0] [0 0 0 0] sage: gale_ryser_theorem([0,0,0],[0,0,0,0], algorithm="ryser") [0 0 0 0] [0 0 0 0] [0 0 0 0]
Check that trac ticket #16638 is fixed:
sage: tests = [([4, 3, 3, 2, 1, 1, 1, 1, 0], [6, 5, 1, 1, 1, 1, 1]), ....: ([4, 4, 3, 3, 1, 1, 0], [5, 5, 2, 2, 1, 1]), ....: ([4, 4, 3, 2, 1, 1], [5, 5, 1, 1, 1, 1, 1, 0, 0]), ....: ([3, 3, 3, 3, 2, 1, 1, 1, 0], [7, 6, 2, 1, 1, 0]), ....: ([3, 3, 3, 1, 1, 0], [4, 4, 1, 1, 1])] sage: for s1, s2 in tests: ....: m = gale_ryser_theorem(s1, s2, algorithm="ryser") ....: ss1 = sorted(map(lambda x : sum(x) , m.rows()), reverse = True) ....: ss2 = sorted(map(lambda x : sum(x) , m.columns()), reverse = True) ....: if ((ss1 != s1) or (ss2 != s2)): ....: print("Error in Ryser algorithm") ....: print(s1, s2)
REFERENCES:
[Ryser63] (1, 2) H. J. Ryser, Combinatorial Mathematics, Carus Monographs, MAA, 1963. [Gale57] (1, 2) D. Gale, A theorem on flows in networks, Pacific J. Math. 7(1957)1073-1082.
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sage.combinat.integer_vector.
is_gale_ryser
(r, s)¶ Tests whether the given sequences satisfy the condition of the Gale-Ryser theorem.
Given a binary matrix \(B\) of dimension \(n\times m\), the vector of row sums is defined as the vector whose \(i^{\mbox{th}}\) component is equal to the sum of the \(i^{\mbox{th}}\) row in \(A\). The vector of column sums is defined similarly.
If, given a binary matrix, these two vectors are easy to compute, the Gale-Ryser theorem lets us decide whether, given two non-negative vectors \(r,s\), there exists a binary matrix whose row/colum sums vectors are \(r\) and \(s\).
This functions answers accordingly.
INPUT:
r
,s
– lists of non-negative integers.
ALGORITHM:
Without loss of generality, we can assume that:
- The two given sequences do not contain any \(0\) ( which would correspond to an empty column/row )
- The two given sequences are ordered in decreasing order (reordering the sequence of row (resp. column) sums amounts to reordering the rows (resp. columns) themselves in the matrix, which does not alter the columns (resp. rows) sums.
We can then assume that \(r\) and \(s\) are partitions (see the corresponding class
Partition
)If \(r^*\) denote the conjugate of \(r\), the Gale-Ryser theorem asserts that a binary Matrix satisfying the constraints exists if and only if \(s\preceq r^*\), where \(\preceq\) denotes the domination order on partitions.
EXAMPLES:
sage: from sage.combinat.integer_vector import is_gale_ryser sage: is_gale_ryser([4,2,2],[3,3,1,1]) True sage: is_gale_ryser([4,2,1,1],[3,3,1,1]) True sage: is_gale_ryser([3,2,1,1],[3,3,1,1]) False
REMARK: In the literature, what we are calling a Gale-Ryser sequence sometimes goes by the (rather generic-sounding) term ‘’realizable sequence’‘.
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sage.combinat.integer_vector.
list2func
(l, default=None)¶ Given a list
l
, return a function that takes in a value \(i\) and return \(l[i]\). If default is not None, then the function will return the default value for out of range \(i\)‘s.EXAMPLES:
sage: f = sage.combinat.integer_vector.list2func([1,2,3]) sage: f(0) 1 sage: f(1) 2 sage: f(2) 3 sage: f(3) Traceback (most recent call last): ... IndexError: list index out of range
sage: f = sage.combinat.integer_vector.list2func([1,2,3], 0) sage: f(2) 3 sage: f(3) 0