# Counting, generating, and manipulating non-negative integer matrices¶

Counting, generating, and manipulating non-negative integer matrices with prescribed row sums and column sums.

AUTHORS:

• Franco Saliola
class sage.combinat.integer_matrices.IntegerMatrices(row_sums, column_sums)¶

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

The class of non-negative integer matrices with prescribed row sums and column sums.

An integer matrix $$m$$ with column sums $$c := (c_1,...,c_k)$$ and row sums $$l := (l_1,...,l_n)$$ where $$c_1+...+c_k$$ is equal to $$l_1+...+l_n$$, is a $$n \times k$$ matrix $$m = (m_{i,j})$$ such that $$m_{1,j}+\dots+m_{n,j} = c_j$$, for all $$j$$ and $$m_{i,1}+\dots+m_{i,k} = l_i$$, for all $$i$$.

EXAMPLES:

There are $$6$$ integer matrices with row sums $$[3,2,2]$$ and column sums $$[2,5]$$:

```sage: from sage.combinat.integer_matrices import IntegerMatrices
sage: IM = IntegerMatrices([3,2,2], [2,5]); IM
Non-negative integer matrices with row sums [3, 2, 2] and column sums [2, 5]
sage: IM.list()
[
[2 1]  [1 2]  [1 2]  [0 3]  [0 3]  [0 3]
[0 2]  [1 1]  [0 2]  [2 0]  [1 1]  [0 2]
[0 2], [0 2], [1 1], [0 2], [1 1], [2 0]
]
sage: IM.cardinality()
6
```
cardinality()¶

The number of integer matrices with the prescribed row sums and columns sums.

EXAMPLES:

```sage: from sage.combinat.integer_matrices import IntegerMatrices
sage: IntegerMatrices([2,5], [3,2,2]).cardinality()
6
sage: IntegerMatrices([1,1,1,1,1], [1,1,1,1,1]).cardinality()
120
sage: IntegerMatrices([2,2,2,2], [2,2,2,2]).cardinality()
282
sage: IntegerMatrices([4], [3]).cardinality()
0
sage: len(IntegerMatrices([0,0], [0]).list())
1
```

This method computes the cardinality using symmetric functions. Below are the same examples, but computed by generating the actual matrices:

```sage: from sage.combinat.integer_matrices import IntegerMatrices
sage: len(IntegerMatrices([2,5], [3,2,2]).list())
6
sage: len(IntegerMatrices([1,1,1,1,1], [1,1,1,1,1]).list())
120
sage: len(IntegerMatrices([2,2,2,2], [2,2,2,2]).list())
282
sage: len(IntegerMatrices([4], [3]).list())
0
sage: len(IntegerMatrices([0], [0]).list())
1
```
column_sums()¶

The column sums of the integer matrices in self.

OUTPUT:

• Composition

EXAMPLES:

```sage: from sage.combinat.integer_matrices import IntegerMatrices
sage: IM = IntegerMatrices([3,2,2], [2,5])
sage: IM.column_sums()
[2, 5]
```
row_sums()¶

The row sums of the integer matrices in self.

OUTPUT:

• Composition

EXAMPLES:

```sage: from sage.combinat.integer_matrices import IntegerMatrices
sage: IM = IntegerMatrices([3,2,2], [2,5])
sage: IM.row_sums()
[3, 2, 2]
```
to_composition(x)¶

The composition corresponding to the integer matrix.

This is the composition obtained by reading the entries of the matrix from left to right along each row, and reading the rows from top to bottom, ignore zeros.

INPUT:

• x – matrix

EXAMPLES:

```sage: from sage.combinat.integer_matrices import IntegerMatrices
sage: IM = IntegerMatrices([3,2,2], [2,5]); IM
Non-negative integer matrices with row sums [3, 2, 2] and column sums [2, 5]
sage: IM.list()
[
[2 1]  [1 2]  [1 2]  [0 3]  [0 3]  [0 3]
[0 2]  [1 1]  [0 2]  [2 0]  [1 1]  [0 2]
[0 2], [0 2], [1 1], [0 2], [1 1], [2 0]
]
sage: for m in IM: print IM.to_composition(m)
[2, 1, 2, 2]
[1, 2, 1, 1, 2]
[1, 2, 2, 1, 1]
[3, 2, 2]
[3, 1, 1, 1, 1]
[3, 2, 2]
```
sage.combinat.integer_matrices.integer_matrices_generator(row_sums, column_sums)¶

Recursively generate the integer matrices with the prescribed row sums and column sums.

INPUT:

• row_sums – list or tuple
• column_sums – list or tuple

OUTPUT:

• an iterator producing a list of lists

EXAMPLES:

```sage: from sage.combinat.integer_matrices import integer_matrices_generator
sage: iter = integer_matrices_generator([3,2,2], [2,5]); iter
<generator object integer_matrices_generator at ...>
sage: for m in iter: print m
[[2, 1], [0, 2], [0, 2]]
[[1, 2], [1, 1], [0, 2]]
[[1, 2], [0, 2], [1, 1]]
[[0, 3], [2, 0], [0, 2]]
[[0, 3], [1, 1], [1, 1]]
[[0, 3], [0, 2], [2, 0]]
```

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