The Blocks class

A Blocks object represents a signed partition of the set \(\{0, \ldots, n - 1\}\).

It is possible to associate to every Bipartition a pair of blocks, Bipartition.left_blocks() and Bipartition.right_blocks(), which determine the Green’s \(\mathscr{L}\)- and \(\mathscr{R}\)-classes of the Bipartition in the monoid of all bipartitions. This is the purpose of this class.

See also Bipartition for more details and context.

Contents

`Blocks`	A `Blocks` object represents a signed partition of the set \(\{0, \ldots, n - 1\}\).
`Blocks.copy`(…)	Copy a Blocks object.
`Blocks.degree`(…)	Return the degree of a blocks object.
`Blocks.is_transverse_block`(…)	Check if a block is a transverse block.
`Blocks.iterator`(…)	Return a const iterator yielding the indices of the blocks.
`Blocks.lookup`(…)	Returns the transverse blocks lookup.
`Blocks.number_of_blocks`(…)	Return the number of blocks in a Blocks object.
`Blocks.rank`(…)	Return the number of transverse blocks.

Full API

class Blocks

__init__(self: Blocks, blocks: list[list[int]]) → None

Constructs a Blocks object from a list of lists of integers.

This function constructs a Blocks object from a list of lists of integers, so that the blocks consisting of negative values are transverse and those consisting of positive values are not.

Parameters:

blocks (list[list[int]]) – the blocks

Raises:

LibsemigroupsError – if the set consisting of the absolute values of the entries in blocks is not \(\{1, \ldots, n\}\) where \(n\) is the maximum such value.
LibsemigroupsError – if 0 is an item in any block.
LibsemigroupsError – if any block is empty.
LibsemigroupsError – if any block contains both negative and positive values.
LibsemigroupsError – if the constructed Blocks object is not valid.

Complexity:

linear in the sum of the sizes of the lists in blocks.

copy(self: Blocks) → Blocks

Copy a Blocks object.

Returns:: A copy.
Return type:: Blocks

degree(self: Blocks) → int

Return the degree of a blocks object. The degree of a Blocks object is the size of the set of which it is a partition, or the size of the blocks used to construct self.

Returns:: The degree of a Blocks object.
Return type:: int

is_transverse_block(self: Blocks, index: int) → bool

Check if a block is a transverse block.

This function returns True if the block with index index is a transverse (or signed) block and it returns False if it is not transverse (or unsigned).

Parameters:: index (int) – the index of a block
Returns:: Whether or not the given block is transverse.
Return type:: bool
Raises:: LibsemigroupsError – if index is not in the range \([0, n)\) where \(n\) is the return value of number_of_blocks.
Complexity:: Constant.

iterator(self: Blocks) → collections.abc.Iterator[int]

Return a const iterator yielding the indices of the blocks.

Returns:: An iterator yielding the indices of blocks.
Return type:: collections.abc.Iterator[int]
Complexity:: Constant.

lookup(self: Blocks) → list[bool]

Returns the transverse blocks lookup.

This function returns the transverse blocks lookup. The value in position i of the returned list is True if the block with index i is transverse and False if it is not.

Returns:: The transverse block lookup list.
Return type:: list[bool]
Complexity:: Constant.

number_of_blocks(self: Blocks) → int

Return the number of blocks in a Blocks object. This function returns the number of parts in the partition that instances of this class represent.

Returns:: The number of blocks.
Return type:: int
Complexity:: At worst \(O(n)\) where \(n\) is degree().

rank(self: Blocks) → int

Return the number of transverse blocks. This function returns the number of True values in lookup().

Returns:: The number of signed (transverse) blocks in self.
Return type:: int
Complexity:: At most linear in the number of blocks.