permualgebra 0.0.1

Calculate the permutations with Python.

Permualgebra

This package allows user to do permutations calculation in terms of Modern Algebra.

Install

pip install permualgebra

Permutation

Definition

Let S be a set of n distinct elements. A permutation of S is a bijection

For example, let S = {1, 2, 3, 4, 5, 6}, define

i	1	2	3	4	5	6
p(i)	6	3	2	4	5	1

Or a permutation can be written in cycle notation, where we often omit the 1-cycles:

this cycle notation is also the way that this package express a permutation.

The length of a cycle is the number of elements of S in that cycle.

The length of a permutation is the number of cycles in that permutation.

Theorem

Every permutation on S = {1, ..., n} can be written as a product of disjoint cycles. i.e. no elements of S is repeated in the cycle description.

import permualgebra as pm
p = pm.Permutation(["3 6 4 2", "1 3 4 6", "5 2 1 3"])
print(p)		# (3 6 4 2)(1 3 4 6)(5 2 1 3)
pSimplify = p.getSimplify()
print(pSimplify)# (1 2 6)(3 5)
p.simplify()
print(p)		# (1 2 6)(3 5)

Suppose we have 2 permutations:

we can compose them as

Note. composition of permutations is not commutative.

This package implements the composition of permutation as multiplication.

import permualgebra as pm

p = pm.Permutation(["1 5", "2 4 6"])
q = pm.Permutation(["1 3 5 4", "2 6"])
p.simplify()
q.simplify()
print((p*q).getSimplify())	# (1 3)(4 5 6)
print((q*p).getSimplify())	# (1 4 2)(3 5)

To Do

• Errors and Exceptions
• More detailed wiki and documentation.