permutation — Permutations of finitely many positive integers

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permutation provides a Permutation class for representing permutations of finitely many positive integers in Python. Supported operations & properties include inverses, (group theoretic) order, parity, composition/multiplication, cycle decomposition, cycle notation, word representation, Lehmer codes, and, of course, use as a callable on integers.

Installation

permutation requires Python 3.10 or higher. Just use pip for Python 3 (You have pip, right?) to install:

python3 -m pip install permutation

Examples

>>> from permutation import Permutation
>>> p = Permutation(2, 1, 4, 5, 3)
>>> p(1)
2
>>> p(3)
4
>>> p(42)
42
>>> p.to_cycles()
[(1, 2), (3, 4, 5)]
>>> print(p)
(1 2)(3 4 5)
>>> print(p.inverse())
(1 2)(3 5 4)
>>> p.degree
5
>>> p.order
6
>>> p.is_even
False
>>> p.lehmer(5)
27
>>> q = Permutation.cycle(1,2,3)
>>> print(p * q)
(2 4 5 3)
>>> print(q * p)
(1 3 4 5)
>>> for p in Permutation.group(3):
...     print(p)
...
1
(1 2)
(2 3)
(1 3 2)
(1 2 3)
(1 3)

API

class permutation.Permutation(*img: int)[source]

A Permutation object represents a permutation of finitely many positive integers, i.e., a bijective function from some integer range \([1,n]\) to itself.

Permutations are hashable and immutable. They can be compared for equality but not for ordering/sorting.

Construction:

`__init__`(*img)	Construct a permutation from a word representation.
`parse`(s)	Parse a permutation written in cycle notation.
`cycle`(*cyc)	Construct a cyclic permutation.
`from_cycles`(*cycles)	Construct the product of cyclic permutations.
`from_lehmer`(x, n)	Calculate the permutation in \(S_n\) with Lehmer code `x`.
`from_left_lehmer`(x)	Calculate the permutation with the given left Lehmer code.
`group`(n)	Generates all permutations in \(S_n\).
`next_permutation`()	Returns the next `Permutation` in left Lehmer code order
`prev_permutation`()	Returns the previous `Permutation` in left Lehmer code order

Operations:

`__call__`(i)	Map an integer through the permutation.
`__mul__`(other)	Multiplication/composition of permutations.
`__pow__`(n)	Power/repeated composition of permutations.
`permute`(xs)	Returns the elements of `xs` reordered according to the permutation.

Properties:

`__str__`()	Convert a `Permutation` to cycle notation.
`__bool__`()	A `Permutation` is true iff it is not the identity.
`inverse`()	Returns the inverse of the permutation.
`degree`	The degree of the permutation.
`order`	The order/period of the permutation.
`is_even`	Whether the permutation is even.
`is_odd`	Whether the permutation is odd, i.e., not even
`sign`	The sign/signature of the permutation.
`isdisjoint`(other)	Tests whether the permutation is disjoint from `other`.
`to_image`([n])	Returns the images of 1 through `n` under the permutation.
`to_cycles`()	Decompose the permutation into a product of disjoint cycles.
`right_inversion_count`([n])	Calculate the right inversion count through degree `n`.
`inversions`()	Calculate the inversion number of the permutation.
`lehmer`(n)	Calculate a Lehmer code for the permutation.
`left_lehmer`()	Calculate the "left Lehmer code" for the permutation.

__bool__() → bool[source]

A Permutation is true iff it is not the identity.

__call__(i: int) → int[source]

Map an integer through the permutation. Values less than 1 are returned unchanged.

Parameters:: i (int)
Returns:: the image of i under the permutation

__init__(*img: int) → None[source]

Construct a permutation from a word representation. The arguments are the images of the integers 1 through some \(n\) under the permutation to construct.

For example, Permutation(5, 4, 3, 6, 1, 2) is the permutation that maps 1 to 5, 2 to 4, 3 to itself, 4 to 6, 5 to 1, and 6 to 2. Permutation() (with no arguments) evaluates to the identity permutation (i.e., the permutation that returns all inputs unchanged).

__mul__(other: Permutation) → Permutation[source]

Multiplication/composition of permutations. p * q returns a Permutation r such that r(x) == p(q(x)) for all integers x.

Parameters:: other (Permutation)
Return type:: Permutation

__pow__(n: int) → Permutation[source]

Power/repeated composition of permutations.

p ** 0 == Permutation()
p ** n == p ** (n - 1) * p
p ** -n == p.inverse() ** n

Parameters:: n (int) – exponent
Return type:: Permutation

__str__() → str[source]

Convert a Permutation to cycle notation. The instance is decomposed into cycles with to_cycles(), each cycle is written as a parenthesized space-separated sequence of integers, and the cycles are concatenated.

str(Permutation()) is "1".

This is the inverse of parse.

>>> str(Permutation(2, 5, 4, 3, 1))
'(1 2 5)(3 4)'

classmethod cycle(*cyc: int) → Permutation[source]

Construct a cyclic permutation. If p = Permutation.cycle(*cyc), then p(cyc[0]) == cyc[1], p(cyc[1]) == cyc[2], etc., and p(cyc[-1]) == cyc[0], with p returning all other values unchanged.

Permutation.cycle() (with no arguments) evaluates to the identity permutation.

Parameters:

cyc – zero or more distinct positive integers

Returns:

the permutation represented by the given cycle

Raises:

ValueError –

if cyc contains a value less than 1
if cyc contains the same value more than once

property degree: int

The degree of the permutation. This is the largest integer that it permutes (does not map to itself), or 0 if there is no such integer (i.e., if the permutation is the identity).

classmethod from_cycles(*cycles: Iterable[int]) → Permutation[source]

Construct the product of cyclic permutations. Each element of cycles is converted to a Permutation with cycle, and the results (which need not be disjoint) are multiplied together. Permutation.from_cycles() (with no arguments) evaluates to the identity permutation.

This is the inverse of to_cycles.

Parameters:

cycles – zero or more iterables of distinct positive integers

Returns:

the Permutation represented by the product of the cycles

Raises:

ValueError –

if any cycle contains a value less than 1
if any cycle contains the same value more than once

classmethod from_left_lehmer(x: int) → Permutation[source]

Calculate the permutation with the given left Lehmer code. This is the inverse of left_lehmer().

Parameters:: x (int) – a nonnegative integer
Returns:: the Permutation with left Lehmer code x
Raises:: ValueError – if x is less than 0

classmethod from_lehmer(x: int, n: int) → Permutation[source]

Calculate the permutation in \(S_n\) with Lehmer code x. This is the permutation at index x (zero-based) in the list of all permutations of degree at most n ordered lexicographically by word representation.

This is the inverse of lehmer.

Parameters:

x (int) – a nonnegative integer
n (int) – the degree of the symmetric group with respect to which x was calculated

Returns:

the Permutation with Lehmer code x

Raises:

ValueError – if x is less than 0 or greater than or equal to the factorial of n

classmethod group(n: int) → Iterator[Permutation][source]

Generates all permutations in \(S_n\). This is the symmetric group of degree n, i.e., all permutations with degree less than or equal to n. The permutations are yielded in ascending order of their left Lehmer codes.

Parameters:: n (int) – a nonnegative integer
Returns:: a generator of all Permutations with degree n or less
Raises:: ValueError – if n is less than 0

inverse() → Permutation[source]

Returns the inverse of the permutation. This is the unique permutation that, when multiplied by the invocant on either the left or the right, produces the identity.

Return type:: Permutation

inversions() → int[source]

Calculate the inversion number of the permutation. This is the number of pairs of numbers which are in the opposite order after applying the permutation. This is also the Kendall tau distance from the identity permutation. This is also the sum of the terms in the Lehmer code when in factorial base.

Added in version 0.2.0.

Returns:: the number of inversions in the permutation
Return type:: int

property is_even: bool

Whether the permutation is even. That is, whether it can be expressed as the product of an even number of transpositions (cycles of length 2).

property is_odd: bool

Whether the permutation is odd, i.e., not even

isdisjoint(other: Permutation) → bool[source]

Tests whether the permutation is disjoint from other. This returns True iff the two permutations do not permute any of the same integers.

Parameters:: other (Permutation) – a permutation to compare against
Return type:: bool

left_lehmer() → int[source]

Calculate the “left Lehmer code” for the permutation. This uses a modified form of Lehmer codes that uses the left inversion count instead of the right inversion count. This modified encoding establishes a degree-independent bijection between permutations and nonnegative integers, with from_left_lehmer() converting values in the opposite direction.

Returns:: the permutation’s left Lehmer code
Return type:: int

lehmer(n: int) → int[source]

Calculate a Lehmer code for the permutation. The Lehmer code is computed with respect to all permutations of degree at most n and evaluates to the zero-based index of the permutation in the list of all such permutations when ordered lexicographically by word representation.

This is the inverse of from_lehmer.

Parameters:: n (int)
Return type:: int
Raises:: ValueError – if n is less than degree

next_permutation() → Permutation[source]

Returns the next Permutation in left Lehmer code order

property order: int

The order/period of the permutation. This is the smallest positive integer \(n\) such that multiplying \(n\) copies of the permutation together produces the identity

classmethod parse(s: str) → Permutation[source]

Parse a permutation written in cycle notation. This is the inverse of __str__.

Parameters:: s (str) – a permutation written in cycle notation
Returns:: the permutation represented by s
Return type:: Permutation
Raises:: ValueError – if s is not valid cycle notation for a permutation

permute(xs: Iterable[T]) → list[T][source]

Returns the elements of xs reordered according to the permutation. Each element at index i is moved to index p(i).

Note that p.permute(range(1, n+1)) == p.inverse().to_image(n) for all integers n greater than or equal to degree.

Changed in version 0.5.0: This method now accepts iterables of any element type and returns a list. (Previously, it only accepted iterables of ints and returned a tuple.)

Parameters:: xs – a sequence of at least degree elements
Returns:: a permuted sequence
Return type:: list
Raises:: ValueError – if len(xs) is less than degree

prev_permutation() → Permutation[source]

Returns the previous Permutation in left Lehmer code order

Raises:: ValueError – if called on the identity Permutation (which has no predecessor)

right_inversion_count(n: int | None = None) → list[int][source]

Calculate the right inversion count through degree n. The result is a list of n elements in which the element at index i corresponds to the number of right inversions for i+1, i.e., the number of values x > i+1 for which p(x) < p(i+1).

Setting n larger than degree causes the resulting list to have trailing zeroes, which become relevant when converting to & from Lehmer codes and factorial base.

Added in version 0.2.0.

Parameters:: n (Optional[int]) – defaults to degree
Return type:: list[int]
Raises:: ValueError – if n is less than degree

property sign: int

The sign/signature of the permutation. This is 1 if the permutation is even, -1 if it is odd.

to_cycles() → list[tuple[int, ...]][source]

Decompose the permutation into a product of disjoint cycles. to_cycles() returns a list of cycles, each one represented as a tuple of integers. Each cycle c is a sub-permutation that maps c[0] to c[1], c[1] to c[2], etc., finally mapping c[-1] back around to c[0]. The product of these cycles is then the original permutation.

Each cycle is at least two elements in length and places its smallest element first. Cycles are ordered by their first elements in increasing order. No two cycles share an element.

When the permutation is the identity, to_cycles() returns an empty list.

This is the inverse of from_cycles.

Returns:: the cycle decomposition of the permutation

to_image(n: int | None = None) → tuple[int, ...][source]

Returns the images of 1 through n under the permutation. If v = p.to_image(), then v[0] == p(1), v[1] == p(2), etc.

When the permutation is the identity, to_image called without an argument returns an empty tuple.

This is the inverse of the constructor.

Parameters:: n (int) – the length of the image to return; defaults to degree
Returns:: the image of 1 through n under the permutation
Return type:: tuple[int, …]
Raises:: ValueError – if n is less than degree

permutation — Permutations of finitely many positive integers

Installation

Examples

API

Indices and tables