from typing import Any, Set, Dict, List, Tuple, Union, Iterable
import symmetria.elements.cycle
import symmetria.elements.cycle_decomposition
from symmetria.elements._base import _Element
__all__ = ["Permutation"]
[docs]
class Permutation(_Element):
r"""The ``Permutation`` class represents an element of the symmetric group as a map, i.e., a bijective function
from a finite set of integer :math:`\{1, ..., n\}`, for some :math:`n \in \mathbb{N}_{>0}`, to itself.
To define a permutation, it is needed to provide a sequence of integers defining the image of the permutation.
For example, to define the permutation :math:`\sigma \in S_3` given by :math:`\sigma(1)=3, \sigma(2)=1`, and
:math:`\sigma (3)=2`, you should write ``Permutation(3, 1, 2)``.
:param image: Set of integers defining the image of the permutation.
:type image: int
:raises ValueError: If there is an integer in the provided image which is not strictly positive.
:raises ValueError: If there is an integer which is strictly greater than the total number of integers.
:raises ValueError: If there are repeated integers.
:example:
>>> from symmetria import Permutation
...
>>> permutation = Permutation(3, 1, 2)
>>> permutation = Permutation(*[3, 1, 2])
>>> permutation = Permutation(*(3, 1, 2))
"""
__slots__ = ["_map", "_domain", "_image"]
def __init__(self, *image: int) -> None:
self._map: Dict[int, int] = self._validate_image(image)
self._domain: Iterable[int] = range(1, len(self._map) + 1)
self._image: Tuple[int] = tuple(image)
@staticmethod
def _validate_image(image: Tuple[int, ...]) -> Dict[int, int]:
"""Private method to check if a set of integers is eligible as image for a permutation.
Recall that, a tuple of integers represent the image of a permutation if the following conditions hold:
- all the integers are strictly positive;
- all the integers are bounded by the total number of integers;
- there are no integer repeated.
"""
_map = {}
for idx, img in enumerate(image):
if isinstance(img, int) is False:
raise ValueError(f"Expected `int` type, but got {type(img)}.")
if img < 1:
raise ValueError(f"Expected all strictly positive values, but got {img}")
elif img > len(image):
raise ValueError(f"The permutation is not injecting on its image. Indeed, {img} is not in the image.")
elif img in _map.values():
raise ValueError(
f"It seems that the permutation is not bijective. Indeed, {img} has two, or more, pre-images."
)
else:
_map[idx + 1] = img
return _map
[docs]
def __bool__(self) -> bool:
"""Check if the permutation is different from the identity permutation.
:return: True if the permutation is different from the identity, False otherwise.
:rtype: bool
:example:
>>> from symmetria import Permutation
...
>>> bool(Permutation(1))
False
>>> bool(Permutation(2, 1, 3))
True
"""
return self != Permutation(*self.domain)
[docs]
def __call__(self, item: Any) -> Any:
"""Call the permutation on the `item` object, i.e., mimic a permutation action on the element `item`.
- If `item` is an integer, it applies the permutation to the integer.
- If `item` is a string, a list or a tuple, it applies the permutation permuting the values using the indeces.
- If `item` is a permutation, it returns the multiplication of the two permutations, i.e., the compositions.
- If `item` is a cycle or a cycle decomposition, it returns the composition in cycle decomposition.
:param item: The object on which the permutation acts.
:type item: Any
:return: The permuted object.
:rtype: Any
:raises AssertionError: If the length of the permutation is greater than the length of `item`, i.e.,
the permutation cannot permute the `item`.
:raises ValueError: If the permutation and the object `item` don't belong to the same Symmetric group.
:raises TypeError: If the `item` is not of a supported type. See list above for supported types.
:example:
>>> from symmetria import Permutation
...
>>> permutation = Permutation(3, 1, 2)
>>> permutation(2)
1
>>> permutation("abc")
'bca'
>>> permutation([1, 2, 3])
[2, 3, 1]
>>> permutation(Permutation(3, 1, 2))
Permutation(2, 3, 1)
"""
if isinstance(item, int):
return self._call_on_integer(idx=item)
elif isinstance(item, (str, List, Tuple)):
if len(self) > len(item):
raise ValueError(f"Not enough object to permute {item} using the permutation {self}.")
return self._call_on_str_list_tuple(original=item)
elif isinstance(item, Permutation):
return self * item
elif isinstance(item, symmetria.elements.cycle.Cycle):
if set(item.domain).issubset(set(self.domain)) is False:
raise ValueError(
f"Cannot compose permutation {self} with cycle {item},"
" because they don't live in the same Symmetric group."
)
return self._call_on_cycle(cycle=item)
elif isinstance(item, symmetria.elements.cycle_decomposition.CycleDecomposition):
if self.domain != item.domain:
raise ValueError(
f"Cannot compose permutation {self} with cycle decomposition {item},"
" because they don't live in the same Symmetric group."
)
return self._call_on_cycle_decomposition(item)
raise TypeError(f"Calling a permutation on {type(item)} is not supported.")
def _call_on_integer(self, idx: int) -> int:
"""Private method for calls on integer."""
return self[idx] if 1 <= idx <= len(self) else idx
def _call_on_str_list_tuple(self, original: Union[str, Tuple, List]) -> Union[str, Tuple, List]:
"""Private method for calls on strings, tuples and lists."""
permuted = list(_ for _ in original)
for idx, w in enumerate(original, 1):
permuted[self._call_on_integer(idx=idx) - 1] = w
if isinstance(original, str):
return "".join(permuted)
elif isinstance(original, Tuple):
return tuple(p for p in permuted)
else:
return permuted
def _call_on_cycle(self, cycle: "Cycle") -> "CycleDecomposition":
"""Private method for calls on cycles."""
permutation = []
for element in self.domain:
if element in cycle:
idx = cycle.elements.index(element)
permutation.append(self[cycle[(idx + 1) % len(cycle)]])
else:
permutation.append(self[element])
return Permutation(*permutation).cycle_decomposition()
def _call_on_cycle_decomposition(self, cycle_decomposition: "CycleDecomposition") -> "CycleDecomposition":
"""Private method for calls on cycle decomposition."""
return Permutation.from_dict(
p={idx: self._map[cycle_decomposition.map[idx]] for idx in self.domain}
).cycle_decomposition()
[docs]
def __eq__(self, other: Any) -> bool:
"""Check if the permutation is equal to `another` object.
:param other: The object to compare with.
:type other: Any
:return: True if the permutation is equal to `other`, i.e., they define the same map. Otherwise, False.
:rtype: bool
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3) == Permutation(1, 2, 3)
True
>>> Permutation(1, 2, 3) == Permutation(3, 2, 1)
False
>>> Permutation(1, 2, 3) == 12
False
"""
if isinstance(other, Permutation):
return self.map == other.map
return False
[docs]
def __getitem__(self, item: int) -> int:
"""Return the value of the permutation at the given index `item`.
In other words, it returns the image of the permutation at point `item`.
.. note:: The index corresponds to the element in the domain of the permutation, i.e.,
the index is a number between 1 and the length of the permutation.
:param item: The index of the permutation.
:type item: int
:return: The value of the permutation at the specified index.
:rtype: int
:raises IndexError: If the index is out of range.
:example:
>>> from symmetria import Permutation
...
>>> permutation = Permutation(2, 3, 1)
>>> for idx in range(1, len(permutation)+1):
... permutation[idx]
2
3
1
"""
return self.map[item]
[docs]
def __int__(self) -> int:
"""Convert the permutation to its integer representation.
:return: The integer representation of the permutation.
:rtype: int
:example:
>>> from symmetria import Permutation
...
>>> int(Permutation(3, 1, 2))
312
>>> int(Permutation(1, 3, 4, 5, 2, 6))
134526
"""
return sum([self[element] * 10 ** (len(self) - element) for element in self.domain])
[docs]
def __len__(self) -> int:
"""Return the length of the permutation, which is the number of elements in its domain.
:return: The length of the permutation.
:rtype: int
:example:
>>> from symmetria import Permutation
...
>>> len(Permutation(1))
1
>>> len(Permutation(3, 1, 2))
3
>>> len(Permutation(1, 3, 4, 5, 2, 6))
6
"""
return len(list(self.domain))
[docs]
def __mul__(self, other: "Permutation") -> "Permutation":
"""Multiply the permutation with another permutation, resulting in a new permutation
that represents the composition of the two permutations.
:param other: The other permutation to multiply with.
:type other: Permutation
:return: The composition of the two permutations.
:rtype: Permutation
:raises ValueError: If the permutations don't live in the same Symmetric group.
:raises TypeError: If the other object is not a Permutation.
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3) * Permutation(3, 2, 1)
Permutation(3, 2, 1)
>>> Permutation(1) * Permutation(1)
Permutation(1)
>>> Permutation(3, 4, 5, 1, 2) * Permutation(3, 5, 1, 2, 4)
Permutation(5, 2, 3, 4, 1)
"""
if isinstance(other, Permutation):
if self.domain != other.domain:
raise ValueError(
f"Cannot compose permutation {self} with permutation {other},"
" because they don't live in the same Symmetric group."
)
return Permutation.from_dict(p={idx: self._map[other._map[idx]] for idx in self.domain})
raise TypeError(f"Product between types `Permutation` and {type(other)} is not implemented.")
[docs]
def __pow__(self, power: int) -> "Permutation":
"""Return the permutation object to the chosen power.
:param power: the exponent for the power operation.
:type power: int
:return: the power of the permutation.
:rtype: Permutation
:raises TypeError: If `power` is not an integer.
:example:
>>> from symmetria import Permutation
...
>>> Permutation(3, 1, 2)**0
Permutation(1, 2, 3)
>>> Permutation(3, 1, 2)**1
Permutation(3, 1, 2)
>>> Permutation(3, 1, 2)**-1
Permutation(2, 3, 1)
>>> Permutation(3, 1, 2)**2
Permutation(2, 3, 1)
"""
if isinstance(power, int) is False:
raise TypeError(f"Power operation for type {type(power)} not supported.")
elif self is False or power == 0:
return Permutation(*list(self.domain))
elif power == 1:
return self
elif power <= -1:
return self.inverse() ** abs(power)
else:
return self * (self ** (power - 1))
[docs]
def __repr__(self) -> str:
r"""Return a string representation of the permutation in the format `Permutation(x, y, z, ...)`,
where :math:`x, y, z, ... \in \mathbb{N}` are the elements of the permutation.
:return: A string representation of the permutation.
:rtype: str
:example:
>>> from symmetria import Permutation
...
>>> Permutation(3, 1, 2).__repr__()
'Permutation(3, 1, 2)'
>>> Permutation(1, 3, 4, 5, 2, 6).__repr__()
'Permutation(1, 3, 4, 5, 2, 6)'
"""
return f"Permutation({', '.join([str(self._map[idx]) for idx in self.domain])})"
[docs]
def __str__(self) -> str:
"""Return a string representation of the permutation in the form of a tuple.
The string representation represents the image of the permutation.
:return: A string representation of the permutation.
:rtype: str
:example:
>>> from symmetria import Permutation
...
>>> print(Permutation(3, 1, 2))
(3, 1, 2)
>>> print(Permutation(1, 3, 4, 5, 2, 6))
(1, 3, 4, 5, 2, 6)
"""
return str(self.image) if len(self.image) > 1 else f"({self.image[0]})"
[docs]
def ascents(self) -> List[int]:
r"""Return the ascents of the permutation.
Recall that an ascent of a permutation :math:`\sigma \in S_n`, where :math:`n \in \mathbb{N}`, is any position
:math:`i<n` such that :math:`\sigma(i) < \sigma(i+1)`.
:return: The ascents of the permutation.
:rtype: List[int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).ascents()
[1, 2]
>>> Permutation(3, 4, 5, 2, 1, 6, 7).ascents()
[1, 2, 5, 6]
>>> Permutation(4, 3, 2, 1).ascents()
[]
"""
return [idx + 1 for idx in range(len(self) - 1) if self.image[idx] < self.image[idx + 1]]
[docs]
def cycle_decomposition(self) -> "CycleDecomposition":
"""Decompose the permutation into its cycle decomposition.
:return: The cycle decomposition of the permutation.
:rtype: CycleDecomposition
:example:
>>> from symmetria import Cycle, CycleDecomposition, Permutation
...
>>> Permutation(1).cycle_decomposition()
CycleDecomposition(Cycle(1))
>>> Permutation(3, 1, 2).cycle_decomposition()
CycleDecomposition(Cycle(1, 3, 2))
>>> Permutation(1, 3, 4, 5, 2, 6).cycle_decomposition()
CycleDecomposition(Cycle(1), Cycle(2, 3, 4, 5), Cycle(6))
"""
cycles, visited = [], set()
for idx in self.domain:
if idx not in visited:
orbit = self.orbit(idx)
cycles.append(symmetria.elements.cycle.Cycle(*orbit))
visited.update(orbit)
return symmetria.elements.cycle_decomposition.CycleDecomposition(*cycles)
[docs]
def cycle_notation(self) -> str:
"""Return a string representing the cycle notation of the permutation.
:return: The cycle notation of the permutation.
:rtype: str
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).cycle_notation()
'(1)'
>>> Permutation(3, 1, 2).cycle_notation()
'(1 3 2)'
>>> Permutation(3, 1, 2, 4, 5, 6).cycle_notation()
'(1 3 2)(4)(5)(6)'
"""
return self.cycle_decomposition().cycle_notation()
[docs]
def cycle_type(self) -> Tuple[int]:
r"""Return the cycle type of the permutation.
Recall that the cycle type of the permutation :math:`\sigma` is a sequence of integer, where
There is a 1 for every fixed point of :math:`\sigma`, a 2 for every transposition, and so on.
.. note:: The resulting tuple is sorted in ascending order.
:return: The cycle type of the permutation.
:rtype: Tuple[int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).cycle_type()
(1,)
>>> Permutation(3, 1, 2).cycle_type()
(3,)
>>> Permutation(3, 1, 2, 4, 5, 6).cycle_type()
(1, 1, 1, 3)
>>> Permutation(1, 4, 5, 7, 3, 2, 6).cycle_type()
(1, 2, 4)
"""
return tuple(sorted(len(cycle) for cycle in self.cycle_decomposition()))
[docs]
def descents(self) -> List[int]:
r"""Return the descents of the permutation.
Recall that a descent of a permutation :math:`\sigma \in S_n`, where :math:`n \in \mathbb{N}`, is any position
:math:`i<n` such that :math:`\sigma(i) > \sigma(i+1)`.
:return: The descents of the permutation.
:rtype: List[int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).descents()
[]
>>> Permutation(3, 4, 5, 2, 1, 6, 7).descents()
[3, 4]
>>> Permutation(4, 3, 2, 1).descents()
[1, 2, 3]
"""
return [idx + 1 for idx in range(len(self) - 1) if self.image[idx] > self.image[idx + 1]]
@property
def domain(self) -> Iterable[int]:
"""Return an iterable containing the elements of the domain of the permutation.
The domain of a permutation is the set of indices for which the permutation is defined.
:return: The domain of the permutation.
:rtype: Iterable[int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).domain
range(1, 2)
>>> Permutation(3, 1, 2).domain
range(1, 4)
>>> Permutation(1, 3, 4, 5, 2, 6).domain
range(1, 7)
"""
return self._domain
[docs]
def equivalent(self, other: Any) -> bool:
"""Check if the permutation is equivalent to another object.
This method is introduced because we can have different representation of the same permutation, e.g., as a
cycle, or as cycle decomposition.
:param other: The object to compare with.
:type other: Any
:return: True if the permutation is equivalent to the other object, False otherwise.
:rtype: bool
:example:
>>> from symmetria import Cycle, CycleDecomposition, Permutation
...
>>> Permutation(1, 2, 3).equivalent(Permutation(1, 2, 3))
True
>>> Permutation(3, 1, 2).equivalent(Cycle(1, 3, 2))
True
>>> cycle_decomp = CycleDecomposition(Cycle(1, 2), Cycle(3, 4))
>>> Permutation(2, 1, 4, 3).equivalent(cycle_decomp)
True
"""
if isinstance(other, Permutation):
return self == other
elif isinstance(other, symmetria.elements.cycle.Cycle):
return self == Permutation.from_cycle(other)
elif isinstance(other, symmetria.elements.cycle_decomposition.CycleDecomposition):
return self == Permutation.from_cycle_decomposition(other)
return False
[docs]
def exceedances(self, weakly: bool = False) -> List[int]:
r"""Return the exceedances of the permutation.
Recall that an exceedance of a permutation :math:`\sigma \in S_n`, where :math:`n \in \mathbb{N}`, is any
position :math:`i \in \{ 1, ..., n\}` where :math:`\sigma(i) > i`. An exceedance is called weakly if
:math:`\sigma(i) \geq i`.
:param weakly: `True` to return the weakly exceedances of the permutation. Default `False`.
:type weakly: bool
:return: The exceedances of the permutation.
:rtype: List[int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).exceedances()
[]
>>> Permutation(1, 2, 3).exceedances(weakly=True)
[1, 2, 3]
>>> Permutation(4, 3, 2, 1).exceedances()
[1, 2]
>>> Permutation(3, 4, 5, 2, 1, 6, 7).exceedances()
[1, 2, 3]
>>> Permutation(3, 4, 5, 2, 1, 6, 7).exceedances(weakly=True)
[1, 2, 3, 6, 7]
"""
if weakly:
return [i for i, p in enumerate(self.image, 1) if p >= i]
return [i for i, p in enumerate(self.image, 1) if p > i]
[docs]
@classmethod
def from_cycle(cls, cycle: "Cycle") -> "Permutation":
"""Return a permutation from a cycle.
In other word, it converts a cycle into a permutation.
:param cycle: A cycle.
:type cycle: Cycle
:return: A permutation equivalent to the given cycle.
:rtype: Permutation
:example:
>>> from symmetria import Cycle, Permutation
...
>>> Permutation.from_cycle(Cycle(1))
Permutation(1)
>>> Permutation.from_cycle(Cycle(1, 2, 3))
Permutation(2, 3, 1)
>>> Permutation.from_cycle(Cycle(3))
Permutation(1, 2, 3)
"""
image = []
cycle_length = len(cycle)
for element in range(1, max(cycle.domain) + 1):
if element in cycle:
idx = cycle.elements.index(element)
image.append(cycle[(idx + 1) % cycle_length])
else:
image.append(element)
return Permutation(*image)
[docs]
@classmethod
def from_cycle_decomposition(cls, cycle_decomposition: "CycleDecomposition") -> "Permutation":
"""Return a permutation from a cycle decomposition.
In other word, it converts a cycle decomposition into a permutation.
:param cycle_decomposition: A cycle decomposition.
:type cycle_decomposition: CycleDecomposition
:return: A permutation equivalent to the given cycle.
:rtype: Permutation
:example:
>>> from symmetria import Cycle, CycleDecomposition, Permutation
...
>>> Permutation.from_cycle_decomposition(CycleDecomposition(Cycle(1)))
Permutation(1)
>>> Permutation.from_cycle_decomposition(CycleDecomposition(Cycle(4, 3), Cycle(1, 2)))
Permutation(2, 1, 4, 3)
"""
return Permutation.from_dict(p=cycle_decomposition.map)
[docs]
@classmethod
def from_dict(cls, p: Dict[int, int]) -> "Permutation":
"""Create a permutation object from a dictionary where keys represent indices and values represent the
images of the indeces.
:param p: A dictionary representing the permutation.
:type p: Dict[int, int]
:return: A permutation created from the dictionary.
:rtype: Permutation
:example:
>>> from symmetria import Permutation
...
>>> Permutation.from_dict({1: 3, 2: 1, 3: 2})
Permutation(3, 1, 2)
>>> Permutation.from_dict({1: 5, 2: 3, 3: 1, 4: 2, 5:4})
Permutation(5, 3, 1, 2, 4)
"""
return Permutation(*[p[idx] for idx in range(1, len(p) + 1)])
@property
def image(self) -> Tuple[int]:
r"""Return the image of the permutation.
For example, consider the permutation :math:`\sigma \in S_3` given by :math:`\sigma(1)=3, \sigma(2)=1`, and
:math:`\sigma (3)=2`, then the image of :math:`\sigma` is :math:`(3, 1, 2)` .
:return: The image of the permutation.
:rtype: Tuple[int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).image
(1, 2, 3)
>>> Permutation(1, 3, 4, 2).image
(1, 3, 4, 2)
>>> Permutation(2, 3, 1, 5, 4).image
(2, 3, 1, 5, 4)
"""
return self._image
[docs]
def inverse(self) -> "Permutation":
r"""Return the inverse of the permutation.
Recall that the inverse of a permutation :math:`\sigma \in S_n`, for some :math:`n \in \mathbb{N}`, is the
the only permutation :math:`\tau \in S_n` such that :math:`\sigma * \tau = \tau * \sigma = id`,
where :math:`id` is the identity permutation.
:return: The inverse of the permutation.
:rtype: Permutation
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).inverse()
Permutation(1, 2, 3)
>>> Permutation(1, 3, 4, 2).inverse()
Permutation(1, 4, 2, 3)
>>> Permutation(2, 3, 1, 5, 4).inverse()
Permutation(3, 1, 2, 5, 4)
"""
return Permutation.from_dict({item: key for key, item in self.map.items()})
[docs]
def inversions(self) -> List[Tuple[int, int]]:
r"""Return the inversions of the permutation.
Recall that an inversion of a permutation :math:`\sigma \in S_n`, for :math:`n \in \mathbb{N}`, is a pair
:math:`(i, j)` of positions (indexes), where the entries of the permutation are in the opposite order, i.e.,
:math:`i<j` but :math:`\sigma(i)>\sigma(j)`.
:return: The inversions of the permutation
:rtype: List[Tuple[int, int]]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).inversions()
[]
>>> Permutation(1, 3, 4, 2).inversions()
[(2, 4), (3, 4)]
>>> Permutation(3, 1, 2, 5, 4).inversions()
[(1, 2), (1, 3), (4, 5)]
"""
inversions, image = [], list(self.image)
min_element = 1
for i, p in enumerate(image, 1):
if p == min_element:
min_element += 1
else:
for j, q in enumerate(image[i:], 1):
if p > q:
inversions.append((i, i + j))
return inversions
[docs]
def is_conjugate(self, other: "Permutation") -> bool:
r"""Check if two permutations are conjugated.
Recall that two permutations :math:`\sigma, \quad \tau \in S_n`, for some :math:`n \in \mathbb{N}`, are said to
be conjugated if there is :math:`\gamma \in S_n` such that :math:`\gamma\sigma\gamma^{-1} = \tau`.
:param other: a permutation
:type other: Permutation
:return: True if self and other are conjugated, False otherwise.
:rtype: bool
:raises TypeError: If `other` is not a permutation.
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).is_conjugate(Permutation(1, 2, 3))
True
>>> Permutation(1, 2, 3).is_conjugate(Permutation(3, 2, 1))
False
>>> Permutation(3, 2, 5, 4, 1).is_conjugate(Permutation(5, 2, 1, 4, 3))
True
"""
if isinstance(other, Permutation) is False:
raise TypeError(f"Method `is_conjugate` not implemented for type {type}.")
return self.cycle_type() == other.cycle_type()
[docs]
def is_derangement(self) -> bool:
r"""Check if the permutation is a derangement.
Recall that a permutation :math:`\sigma` is called a derangement if it has no fixed points, i.e.,
:math:`\sigma(x) \neq x` for every :math:`x` in the permutation domain.
:return: True if the permutation is a derangement, False otherwise.
:rtype: bool
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).is_derangement()
False
>>> Permutation(3, 1, 2).is_derangement()
True
>>> Permutation(1, 3, 4, 5, 2, 6).is_derangement()
False
"""
for idx in self.domain:
if self(idx) == idx:
return False
return True
[docs]
def is_even(self) -> bool:
"""Check if the permutation is even.
Recall that a permutation is said to be even if it can be expressed as the product of an even number of
transpositions.
:return: True if the permutation is even, False otherwise.
:rtype: bool
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).is_even()
True
>>> Permutation(2, 1).is_even()
False
>>> Permutation(2, 1, 3).is_even()
False
>>> Permutation(2, 3, 4, 5, 6, 1).is_even()
False
"""
return self.sgn() == 1
[docs]
def is_odd(self) -> bool:
"""Check if the permutation is odd.
Recall that a permutation is said to be odd if it can be expressed as the product of an odd number of
transpositions.
:return: True if the permutation is odd, False otherwise.
:rtype: bool
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).is_odd()
False
>>> Permutation(2, 1).is_odd()
True
>>> Permutation(2, 1, 3).is_odd()
True
>>> Permutation(2, 3, 4, 5, 6, 1).is_odd()
True
"""
return self.sgn() == -1
[docs]
def is_regular(self) -> bool:
"""Check if the permutation is regular.
Recall that a permutation is said regular if all cycles in its cycle decomposition have the same length.
:return: True if the permutation is regular, False otherwise.
:rtype: bool
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).is_regular()
True
>>> Permutation(2, 1).is_regular()
True
>>> Permutation(2, 1, 3).is_regular()
False
"""
cycle_decomposition = self.cycle_decomposition()
return all(len(cycle) == len(cycle_decomposition[0]) for cycle in cycle_decomposition)
@property
def map(self) -> Dict[int, int]:
"""Return a dictionary representing the mapping of the permutation.
The keys of the dictionary are indices, while the values are the corresponding elements after permutation.
:return: The mapping of the permutation.
:rtype: Dict[int, int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).map
{1: 1}
>>> Permutation(3, 1, 2).map
{1: 3, 2: 1, 3: 2}
"""
return self._map
[docs]
def one_line_notation(self) -> str:
r"""Return a string representation of the permutation in the one-line notation, i.e., in the form
:math:`\sigma(x_1)\sigma(x_2)...\sigma(x_n)`, where :math:`\sigma` is a permutation and :math:`x_1, ..., x_n`
are the elements permuted by :math:`\sigma`.
:return: The one-line notation of the permutation.
:rtype: str
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).one_line_notation()
'1'
>>> Permutation(3, 1, 2).one_line_notation()
'312'
>>> Permutation(1, 3, 4, 5, 2, 6).one_line_notation()
'134526'
"""
return str(int(self))
[docs]
def orbit(self, item: Any) -> List[Any]:
r"""Compute the orbit of `item` object under the action of the cycle.
Recall that the orbit of the action of a permutation :math:`\sigma` on an element x is given by the set
.. math:: \{ \sigma^n(x): n \in \mathbb{N}\}
:param item: The initial element or iterable to compute the orbit for.
:type item: Any
:return: The orbit of the specified element under the permutation.
:rtype: List[Any]
:example:
>>> from symmetria import Cycle, CycleDecomposition, Permutation
...
>>> permutation = Permutation(3, 1, 2)
>>> permutation.orbit(1)
[1, 3, 2]
>>> permutation.orbit([1, 2, 3])
[[1, 2, 3], [2, 3, 1], [3, 1, 2]]
>>> permutation.orbit("abc")
['abc', 'bca', 'cab']
>>> permutation.orbit(Permutation(3, 1, 2))
[Permutation(3, 1, 2), Permutation(2, 3, 1), Permutation(1, 2, 3)]
"""
if isinstance(item, symmetria.elements.cycle.Cycle):
item = item.cycle_decomposition()
orbit = [item]
next_element = self(item)
while next_element != item:
orbit.append(next_element)
next_element = self(next_element)
return orbit
[docs]
def order(self) -> int:
r"""Return the order of the permutation.
Recall that the order of a permutation :math:`\sigma` is the smallest positive integer :math:`n \in \mathbb{N}`
such that :math:`\sigma^n = id`, where :math:`id` is the identity permutation.
:return: The order of the permutation.
:rtype: int
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).order()
1
>>> Permutation(3, 1, 2).order()
3
>>> Permutation(1, 3, 4, 5, 2, 6).order()
4
"""
return self.cycle_decomposition().order()
[docs]
def records(self) -> List[int]:
r"""Return the records of the permutation.
Recall that a record of a permutation :math:`\sigma \in S_n`, where :math:`n \in \mathbb{N}`, is a position
:math:`i \in \{1, ..., n\}` such that is either :math:`i=1` or :math:`\sigma(j) < \sigma(i)`
for all :math:`j<i`.
.. note:: There are definitions of records in the literature where the first index is not considered as a
record.
:return: The records of the permutation.
:rtype: List[int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1, 2, 3).records()
[1, 2, 3]
>>> Permutation(3, 1, 2).records()
[1]
>>> Permutation(1, 3, 4, 5, 2, 6).records()
[1, 2, 3, 4, 6]
"""
records = [1]
tmp_max = self[1]
for i in self.domain:
if self[i] > tmp_max:
records.append(i)
tmp_max = self[i]
return records
[docs]
def sgn(self) -> int:
r"""Return the sign of the permutation.
Recall that the sign, signature, or signum of a permutation :math:`\sigma` is defined as +1 if :math:`\sigma`
is even, and -1 if :math:`\sigma` is odd.
:return: 1 if the permutation is even, -1 if the permutation is odd.
:rtype: int
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).sgn()
1
>>> Permutation(2, 1).sgn()
-1
>>> Permutation(2, 3, 4, 5, 6, 1).sgn()
-1
"""
return self.cycle_decomposition().sgn()
[docs]
def support(self) -> Set[int]:
r"""Return a set containing the indices in the domain of the permutation whose images are different from their
respective indices, i.e., the set of :math:`n` in the permutation domain which are not mapped to itself.
:return: The support set of the permutation.
:rtype: Set[int]
:example:
>>> from symmetria import Permutation
...
>>> Permutation(1).support()
set()
>>> Permutation(3, 1, 2).support()
{1, 2, 3}
>>> Permutation(1, 3, 4, 5, 2, 6).support()
{2, 3, 4, 5}
"""
return {idx for idx in self.domain if self(idx) != idx}
