The Perm16 class

Class representing permutations.

SIMD accelerated class Perm16 representing permutations on up to 16 points.

This class belongs to the hpcombi subpackage of libsemigroups_pybind11.

The functionality described on this page is only available if LIBSEMIGROUPS_HPCOMBI_ENABLED is True.

Perm16 inherits from Transf16.

Contents

`Perm16`	Class representing permutations.
`Perm16.copy`(…)	Copy a `Perm16`.
`Perm16.cycles_partition`(…)	Returns the set partition of the cycles of a permutation.
`Perm16.elementary_transposition`(…)	Returns the elementary transposition swapping i and i + 1.
`Perm16.inverse`(…)	Returns the inverse permutation.
`Perm16.inverse_arr`(…)	Returns the inverse permutation.
`Perm16.inverse_cycl`(…)	Returns the inverse permutation.
`Perm16.inverse_find`(…)	Returns the inverse permutation.
`Perm16.inverse_pow`(…)	Returns the inverse permutation.
`Perm16.inverse_ref`(…)	Returns the inverse permutation.
`Perm16.inverse_sort`(…)	Returns the inverse permutation.
`Perm16.left_weak_leq`(…)	Compare two permutations using the left weak order.
`Perm16.left_weak_leq_length`(…)	Compare two permutations using the left weak order.
`Perm16.left_weak_leq_ref`(…)	Compare two permutations using the left weak order.
`Perm16.lehmer`(…)	Returns the Lehmer code of a permutation.
`Perm16.lehmer_arr`(…)	Returns the Lehmer code of a permutation.
`Perm16.lehmer_ref`(…)	Returns the Lehmer code of a permutation.
`Perm16.length`(…)	Returns the Coxeter length of a permutation.
`Perm16.length_arr`(…)	Returns the Coxeter length of a permutation.
`Perm16.length_ref`(…)	Returns the Coxeter length of a permutation.
`Perm16.nb_cycles`(…)	Returns the number of cycles of a permutation.
`Perm16.nb_cycles_ref`(…)	Returns the number of cycles of a permutation.
`Perm16.nb_cycles_unroll`(…)	Returns the number of cycles of a permutation.
`Perm16.nb_descents`(…)	Returns the number of descent of a permutation.
`Perm16.nb_descents_ref`(…)	Returns the number of descent of a permutation.
`Perm16.one`(…)	Returns the identity permutation.
`Perm16.unrankSJT`(…)	Returns the r-th permutation of size `16` in the Steinhaus–Johnson–Trotter order.
`Perm16.validate`(…)	Check whether or not a `Perm16` is well-defined.

Full API

class hpcombi.Perm16

__init__(*args, **kwargs)

Overloaded function.

__init__(self: Perm16) → None

Default constructor.

Constructs a Perm16 object with its entries uninitialized. This means there is no guarantee about the values in the constructed object.

>>> from libsemigroups_pybind11.hpcombi import Perm16
>>> x = Perm16()

__init__(self: Perm16, img: list[int]) → None

Construct a Perm16 from a list of images.

This function constructs a Perm16 from the list img of its entries. If the length of img is less than 16, then the constructed Perm16 is padded with fixed points at the end.

Parameters:

img (list[int]) – The list of images.

Raises:

LibsemigroupsError – if any value in img is in the range \([16, 256)\).
LibsemigroupsError – if the length of img exceeds 16.
LibsemigroupsError – if there are repeated values in img.
TypeError – if any value in img is larger than 255.

>>> from libsemigroups_pybind11.hpcombi import Perm16
>>> Perm16([1, 3, 2, 0])
Perm16([ 1, 3, 2, 0, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15])

__init__(self: Perm16, n: int) → None

Construct a permutation from its 64 bits compressed.

This function constructs a Perm16 from its integer representation n.

Parameters:: n (int) – the integer representation.
Raises:: TypeError – if n is not in the range \([0, 2 ^{64})\).

Warning

For most values of n the returned object is not a valid permutation.

>>> from libsemigroups_pybind11.hpcombi import Perm16
>>> Perm16(17863200008537477760)
Perm16([ 0, 2, 1, 4, 3, 5, 6, 7, 8, 9,10,11,12,13,14,15])
>>> int(Perm16([0, 2, 1, 4, 3]))
17863200008537477760

copy(self: Perm16) → Perm16

Copy a Perm16.

Returns:: A copy of the argument.
Return type:: Perm16

>>> from libsemigroups_pybind11.hpcombi import Perm16
>>> x = Perm16([1, 2, 3, 4, 0])
>>> x.copy() is not x
True
>>> x.copy() == x
True

cycles_partition(self: Perm16) → Vect16

Returns the set partition of the cycles of a permutation.

This function returns the set partition of the cycles of a permutation, represented the vector \(v\) where \(v[i]\) contains the smallest element in the cycle of \(v\) in self.

Returns:: The set partition of the cycles of the permutation.
Return type:: Vect16

>>> from libsemigroups_pybind11.hpcombi import Perm16
>>> x = Perm16([1,2,3,6,0,5,4,7,8,9,10,11,12,15,14,13])
>>> x.cycles_partition()
Vect16([ 0, 0, 0, 0, 0, 5, 0, 7, 8, 9,10,11,12,13,14,13])

static elementary_transposition(i: int) → Perm16

Returns the elementary transposition swapping i and i + 1.

This function returns the elementary transposition swapping i and i + 1. for every value i in \([0, 15)\).

Parameters:: i (int) – the minimum value to be transposed.
Returns:: The transposition \((i\ i + 1)\).
Return type:: Perm16
Raises:: LibsemigroupsError – if any value in i is greater than or equal to 15.

>>> from libsemigroups_pybind11.hpcombi import Perm16
>>> Perm16.elementary_transposition(2)
Perm16([ 0, 1, 3, 2, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15])

inverse(self: Perm16) → Perm16