Discrete helpers/sig perm

Template:Dh-link This is an extension of the class Perm with added negators.
Signed permutations of length $n$ are the $2^n\cdot n!$ elements of the Template:W of dimension $n$. (See e.g. the full octahedral group for the 48 permutations of the cube.)

This class has been created to model transformations between Boolean functions in the same clan. (See Studies of Euler diagrams/transformations.)

import numpy as np

from discretehelpers.sig_perm import SigPerm
from discretehelpers.perm import Perm
from discretehelpers.binv import Binv


sp = SigPerm(sequence=[3, ~1, ~0, 2])

matrix = [
    [ 0,  0, -1,  0],
    [ 0, -1,  0,  0],
    [ 0,  0,  0,  1],
    [ 1,  0,  0,  0]
]

assert sp == SigPerm(matrix=matrix)

assert sp == SigPerm(pair=[3, 11])
assert sp == SigPerm(valneg_index=3, perm_index=11)
assert sp == SigPerm(valneg=[1, 1, 0, 0], perm=[3, 1, 0, 2])
assert sp == SigPerm(valneg={0, 1}, perm=[3, 1, 0, 2, 4, 5, 6])

assert sp == SigPerm(keyneg_index=6, perm_index=11)
assert sp == SigPerm(keyneg=[0, 1, 1, 0], perm=[3, 1, 0, 2])
assert sp == SigPerm(keyneg={1, 2}, perm=[3, 1, 0, 2, 4, 5, 6])

assert sp.pair == (3, 11)
assert (sp.keyneg_index, sp.valneg_index, sp.perm_index) == (6, 3, 11)

assert sp.length == 4

assert sp.sequence() == sp.sequence(4) == [3, ~1, ~0, 2] == [3, -2, -1, 2]
assert sp.sequence_string() == sp.sequence_string(4) == '[3, ~1, ~0, 2]'

assert sp.binv == Binv('1100')
assert sp.perm == Perm([3, 1, 0, 2])

assert np.array_equal(sp.matrix(), matrix)

Template:Collapsible START Template:Dh-link The pattern of negations can be described in two different ways:

• valneg: which values are negated (or which rows of the permutations matrix)
• keyneg: which places are negated (or which columns of the permutations matrix)

When a signed permutation is denoted by a pair (m, n), it is usually the valneg index m and the permutation index n.
But there are cases, where the keyneg index is the better choice. (See here and here.)
The keyneg index is often shown in a square.
E.g. the signed permutation (~3, 1, ~2, 0) has keyneg index , valneg index and permutation index . Template:Collapsible END

assert sp.inverse == SigPerm(sequence=[~2, ~1, 3, 0])
assert sp.inverse.pair == (6, 19)

inverse_matrix = [
    [ 0,  0,  0,  1],
    [ 0, -1,  0,  0],
    [-1,  0,  0,  0],
    [ 0,  0,  1,  0]
]

assert sp * sp.inverse == sp.inverse * sp == SigPerm(sequence=[]) == Perm()

assert np.array_equal(
    np.dot(matrix, inverse_matrix), 
    np.eye(4)
)

This is a subclass of Perm, making the instances comparable.

from discretehelpers.sig_perm import SigPerm
from discretehelpers.perm import Perm


assert SigPerm(sequence=[3, 2, 0, 1]) == Perm([3, 2, 0, 1])
assert SigPerm(sequence=[3, ~2, 0, 1]) != Perm([3, 2, 0, 1])

Schoute permutation

The property schoute_perm is inherited from Perm. (See there.) These are signed Schoute permutations.

name	a	b
pair	(6, 3)	(5, 4)
sequence	[~2, 0, ~1]	[1, ~2, ~0]
cube permutation

from discretehelpers.sig_perm import SigPerm
from discretehelpers.perm import Perm


a = SigPerm(pair=[6, 3])
b = SigPerm(pair=[5, 4])

assert a.inverse == b

a.sequence_string() == '[~2, 0, ~1]'
b.sequence_string() == '[1, ~2, ~0]'

a.schoute_perm == Perm([6, 2, 7, 3, 4, 0, 5, 1], perilen=8)
b.schoute_perm == Perm([5, 7, 1, 3, 4, 6, 0, 2], perilen=8)

Template:Rdrdo parity

Like for normal permutations, the Template:W corresponds to the determinant of the matrix. (But the metribute inherited from Perm is wrong.)