permuta 1.3.0

A comprehensive high performance permutation library.

Permuta is a Python library for working with perms (short for permutations), patterns, and mesh patterns.

Installing

To install Permuta on your system, run:

pip install permuta

It is also possible to install Permuta in development mode to work on the source code, in which case you run the following after cloning the repository:

./setup.py develop

To run the unit tests:

pip install -r test_requirements.txt
./setup.py test

Using Permuta

Once you’ve installed Permuta, it can be imported by a Python script or an interactive Python session, just like any other Python library:

>>> from permuta import *

Importing * from it supplies you with the ‘Perm’ and ‘PermSet’ classes along with the ‘AvoidanceClass’ class (with alias ‘Av’) for generating perms avoiding a set of patterns. It also gives you the ‘MeshPatt’ class and some other submodules which we will not discuss in this readme.

Creating a single perm

Permutations are zero-based in Permuta and can be created using any iterable.

>>> Perm()  # Empty perm
Perm(())
>>> Perm([])  # Another empty perm
Perm(())
>>> Perm((0, 1, 2, 3)) # The zero-based version of 1234
Perm((0, 1, 2, 3))
>>> Perm((2, 1, 3)) # Warning: it will initialise with any iterable
Perm((2, 1, 3))
>>> Perm((2, 1, 3), check=True) # If you are unsure, you can check
Traceback (most recent call last):
    ...
ValueError: Element out of range: 3
>>> Perm((4, 2, 3, 0, 0), check=True)
Traceback (most recent call last):
    ...
ValueError: Duplicate element: 0
>>> Perm("123", check=True)
Traceback (most recent call last):
    ...
TypeError: ''1'' object is not an integer

Permutations can also be created using some specific class methods.

>>> Perm.from_string("201")  # strings
Perm((2, 0, 1))
>>> Perm.one_based((1, 3, 2, 4)) # one-based iterable of integers
Perm((0, 2, 1, 3))
>>> Perm.to_standard("a2gsv3") # standardising any iterable using '<'
Perm((2, 0, 3, 4, 5, 1))
>>> Perm.from_integer(210) # an integer between 0 and 9876543210
Perm((2, 1, 0))
>>> Perm.from_integer(321) # any integer given is standardised
Perm((2, 1, 0))
>>> Perm.from_integer(201)
Perm((2, 0, 1))

Printing perms gives zero-based strings.

>>> print(Perm(()))
ε
>>> print(Perm((2, 1, 0)))
210
>>> print(Perm((6, 2, 10, 9, 3, 8, 0, 1, 5, 11, 4, 7)))
62(10)938015(11)47

The avoids, contains, and occurrence methods enable working with patterns:

>>> p = Perm((0,2,1,3))
>>> p.contains(Perm((2, 1, 0)))
False
>>> p.avoids(Perm((0, 1)))
False
>>> list(p.occurrences_of(Perm((1, 0))))
[(1, 2)]
>>> list(Perm((0, 1)).occurrences_in(p))
[(0, 1), (0, 2), (0, 3), (1, 3), (2, 3)]

The basic symmetries are implemented:

>>> [p.reverse(), p.complement(), p.inverse()]
[Perm((3, 1, 2, 0)), Perm((3, 1, 2, 0)), Perm((0, 2, 1, 3))]

To take direct sums and skew sums we use + and -:

>>> q = Perm((0, 1, 2, 3, 4))
>>> p + q
Perm((0, 2, 1, 3, 4, 5, 6, 7, 8))
>>> p - q
Perm((5, 7, 6, 8, 0, 1, 2, 3, 4))

There are numerous practical methods available:

>>> list(p.fixed_points())
[0, 3]
>>> list(p.ascents())
[0, 2]
>>> list(p.descents())
[1]
>>> list(p.inversions())
[(1, 2)]
>>> p.major_index()
2

Creating a perm class

You might want the set of all perms:

>>> all_perms = PermSet()
>>> print(all_perms)
<The set of all perms>

Perm classes can be specified with a basis:

>>> basis = [Perm((1, 0, 2)), Perm((1, 2, 0))]
>>> basis
[Perm((1, 0, 2)), Perm((1, 2, 0))]
>>> perm_class = Av(basis)
>>> perm_class
Av((Perm((1, 0, 2)), Perm((1, 2, 0))))

When a basis consists of a single element you can pass it directly to Av:

>>> q = Perm((1,0))
>>> len(Av(q).of_length(100))
1

You can ask whether a perm belongs to the perm class:

>>> Perm((3, 2, 1, 0)) in perm_class
True
>>> Perm((0, 2, 1, 3)) in perm_class
False

You can get the n-th perm of the class or iterate:

>>> sorted([perm_class[n] for n in range(8)])
[Perm(()), Perm((0,)), Perm((0, 1)), Perm((1, 0)), Perm((0, 1, 2)), Perm((0, 2, 1)), Perm((2, 0, 1)), Perm((2, 1, 0))]
>>> perm_class_iter = iter(perm_class)
>>> sorted([next(perm_class_iter) for _ in range(8)])
[Perm(()), Perm((0,)), Perm((0, 1)), Perm((1, 0)), Perm((0, 1, 2)), Perm((0, 2, 1)), Perm((2, 0, 1)), Perm((2, 1, 0))]

(BEWARE: Lexicographic order is not guaranteed at the moment!)

The subset of a perm class where the perms are a specific length

You can define a subset of perms of a specific length in the perm class:

>>> perm_class_14 = perm_class.of_length(14)
>>> perm_class_14
Av((Perm((1, 0, 2)), Perm((1, 2, 0)))).of_length(14)

You can ask for the size of the subset because it is guaranteed to be finite:

>>> len(perm_class_14)
8192

The iterating and containment functionality is the same as with perm_class, but indexing has yet to be implemented:

>>> Perm((2, 1, 0)) in perm_class_14
False
>>> Perm((0, 13, 1, 12, 2, 3, 4, 11, 5, 10, 6, 7, 8, 9)) in perm_class_14
True
>>> Perm(range(10)) - Perm(range(4)) in perm_class_14
False
>>> next(iter(perm_class_14)) in perm_class_14
True

License

BSD-3: see the LICENSE file.