Permutation notation

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This article examines different notations for the composition of permutations with each other and with vectors.

Permutations are bijections from a set to itself, and the set does not need to have an order.
But they can also be described as operations that move things from places to other places — which is the natural mental image when the permutation is e.g. a rotation of a cube.
In the end it does not matter what kind of mental image is used to understand what a permutation is or does, as long as there is no ambiguity what result a formula will yield.
When a permutation is interpreted as moving objects from places to other places, there are two ways to describe it.

Visualisations of as an active permutation
Arrow diagram of
5-el perm; active; only P as arrow diagram.svg
Matrix representations of
Left (prefix)
Right (postfix)
Map 5-el perm; active; only P as matrix; prefix; map.svg 5-el perm; active; only P as matrix; postfix; map.svg
Cycle 5-el perm; active; only P as matrix; prefix; cycle.svg 5-el perm; active; only P as matrix; postfix; cycle.svg

There are two ways to assign a matrix to a permutation, corresponding to the two ways of multiplying permutations.

There are two ways to draw arrows in the chosen matrix, one similar to two-line and the other to cycle notation. (The former is used in the blue boxes 14 and 15, the latter in the rest of the article.)

The usual way is as an active permutation or map or substitution:

moves an object from place to place . In the arrow diagram the one-line notation denotes where the arrows go.
The result of applying on a vector is .

But may also be represented by the result of applying it to the natural order. This is called a passive permutation[1] or (re)arrangement or ordering:

replaces an object in position by that in position . In the arrow diagram the one-line notation denotes where the arrows come from.
In this case the result of applying on a vector is .

Confusing these two interpretations will lead to confusing permutations with their inverses.
Active and passive transformations seem to be a related concept.


Composition of permutations is associative, but not commutative.

With prefix notation or left action .

can be thought of as , and as
This notation corresponds to the usual way of writing function composition.

With postfix notation or right action .

can be thought of as , and as (image of) .
This notation is common in group theory.
(In the Python examples in this article (P*Q)(2) = 2^P^Q = Q(P(2)). The result is R(2) = 4).

In its graphics this article shows all possible interpretations, including the passive ones.
But to avoid confusion the accompanying calculations use only active right notation, which is also used by SymPy and Wolfram.


Notations[edit]

Active[edit]

If permuting by gives the permutation is active.

Active left[edit]

Active right[edit]

AR

If the notation should be seen as active right. *

means and can be written .      can be written .

Permuting by may be written in different ways:      (Compare box 12a.)

The abbreviation with the dot corresponds to the function permute in Wolfram. With two permutations it means:

Passive[edit]

If permuting by gives the permutation is passive.

Passive left should be seen as a misunderstanding of active right, and passive right should be seen as a misunderstanding of active left. *


*   (Unless permuting a vector proofs otherwise.)

Examples with 5-element permutations[edit]

Permutations can be combined with:

  • other permutations (boxes 1 to 11)
  • vectors of the same length with arbitrary elements (green boxes 12 and 16)
  • vectors of arbitrary length with elements from the same range,
    i.e. maps with the same codomain as the permutation, but bijectivity not required (blue boxes 14 and 15)
  • scalars from the same range (red boxes 13 and 17, compare blue box 14)

The variables used in the main examples are:

  • the permutations and
  • the vector (in the graphics simply )
  • the vector

In the Wolfram calculations the 1-based equivalents are used.

Permutations of four elements are identified by their index numbers in RevCoLex order, i.e. as the -th finite permutation.
is denoted where it appears among other permutations of only four elements. (See box 19, compare with box 1.)
(Equivalents for the permutations of five elements would be , and [2], but only the letters are used.)


Active vs. passive[edit]

A simple rule to avoid passive permutations: If the rows of a permutation matrix are labelled, the cycles must go clockwise (AR). If the columns are labelled, the cycles must go anticlockwise (AL).

Left vs. right[edit]

Box 2 Ambiguity of with active permutations
Left (AL):     Right (AR):    
5-el perm; active; S.svg
5-el perm; active; composition; S; prefix.svg
5-el perm; active; composition; R; postfix.svg
5-el perm; active; R.svg

The following examples show with arrow diagrams what the composed permutations actually do, and how this can be interpreted as a product in two different ways, corresponding to two different matrix multiplications.

The two arrangements of matrices are symmetric to each other, including the direction of the arrows in the matrices.
In other terms: The matrix image on the left and on the right show exactly the same thing in a different way.
The arrows in both arrangements of matrices are the same as in the arrow diagram to their left.

In prefix notation (left action) means . In postfix notation (right action) means .
Obviously and are just different ways to say the same thing.

Active[edit]


Box 7
First doing and then doing is the same as doing .
AL:    
AR:    

Passive[edit]


Vectors, scalars and non-bijective maps[edit]

Active[edit]

Box 12 Permuting a vector
5-el perm; active; R.svg
5-el perm; active; vector; P; prefix.svg
5-el perm; active; vector; Q; prefix.svg
5-el perm; active; vector; R; prefix.svg
5-el perm; active; vector; P; postfix.svg
5-el perm; active; vector; Q; postfix.svg
5-el perm; active; vector; R; postfix.svg

Passive[edit]

Examples with permutations of the square[edit]

The dihedral group Dih4 is the group of symmetries of the square. It has 8 elements and is not abelian - therefore order of operation matters.

Active right (Cayley tables)[edit]

In Cayley graphs postfix notation is common. One may find it more natural here, because this way the product read from left to right corresponds to the path from the identity vertex to the vertex of the permutation. (E.g. the path from to in the example below is first through the arrow and then through the edge .) A source that uses this convention is Visual Group Theory by Nathan Carter.

Active left (linear maps)[edit]

This section uses prefix notation, because here the permutations are connected to 2×2 signed permutation matrices describing linear maps. E.g. the matrices of , and are rotation matrices. Linear maps and their matrices are usually concatenated like funcions, i.e. in prefix notation.

permutations as linear maps
Square permutation 0 0.svg Square permutation 1 0.svg Square permutation 2 0.svg Square permutation 3 0.svg Square permutation 0 1.svg Square permutation 1 1.svg Square permutation 2 1.svg Square permutation 3 1.svg

Active vs. passive[edit]

Left vs. right[edit]

Active[edit]

Box 20
AL:    
AR:    

Passive[edit]

Languages[edit]

Wolfram[edit]

as permutation matrix and arrow diagram in Wolfram Alpha

Wolfram Alpha displays permutations in a way that resembles PL representation in this article,
which – if only permutation concatenation is concerned – is the same as AR. (See boxes 1 and 19.)

is shown as , but in the arrow diagram and the permutation matrix the arrow from 1 goes to 3 — not to 2.

Wolfram does not accept the element 0, so the examples are converted to 1-based permutations:

P = (1243)(5)   = 2 4 1 3 5  http://www.wolframalpha.com/input/?i=permutation+(1+2+4+3)(5)
Q = (15)(24)(3) = 5 4 3 2 1  http://www.wolframalpha.com/input/?i=permutation+(1+5)(2+4)(3)
R = (1435)(2)   = 4 2 5 3 1  http://www.wolframalpha.com/input/?i=permutation+(1+4+3+5)(2)
S = (1523)(4)   = 5 3 1 4 2  http://www.wolframalpha.com/input/?i=permutation+(1+5+2+3)(4)

A blogpost from 2011 shows that back then the list notation with curly braces was the inverse of the one-line notation.
But in the calculations shown in this article (done in Mathematica Online in December 2016) the permutation p = {2,4,1,3,5} corresponds to Cycles[{{1,2,4,3}}].
Apparently in the meantime a notation people found confusing was dropped in favour of a more mainstream one.

This article uses three binary Wolfram functions:

PermutationProduct[a, b, c] gives the product of permutations a, b, c.     (box 5b)

PermutationReplace[expr, perm] replaces each part in expr by its image under the permutation perm.     (boxes 13a, 15a, 17a)

Permute[l, p] permutes list l according to permutation p.     (boxes 12a, 16a)


SymPy[edit]

http://docs.sympy.org/dev/modules/combinatorics/permutations.html

The composite of two permutations p*q means first apply p, then q, so i^(p*q) = (i^p)^q which is i^p^q according to Python precedence rules.

SymPy has two operators for permutations: Multiplication (*) and what could be called exponentiation (^). The latter seems to be intended only to access elements of a permutation, but can be used to combine two combinations. The result of a^b is the same as ~b*a*b (see Python script), which is a permutation in the same conjugacy class.

Permutations

Example:   


Example:   
      (details on file page)

Example:   



Sources[edit]

There appear to be no sources that actually use the passive interpretation of permutations. So the sources below are active left and active right.

Left[edit]

Aigner 2007[edit]

We read a product always from right to left, thus for

,   ,

we have

  and   .


We call the word representation of .

Aigner, Martin (2007). A course in enumeration. Berlin New York: Springer. ISBN 3642072534.Word Representation (p. 27 ff)

Right[edit]

Knuth 1973[edit]

In TAoCP Knuth states that in his book permutations are always multiplied from left to right, and gives the following example:

The left permutation sees the argument first:

Notice that the image of 1 under is , etc.

SymPy: (1^a)^b = 5^b = 0

This is not to be confused with right to left multiplication of permutations, i.e. the way functions are composed:

[...] traditional functional notation, in which one writes , makes it natural to think that should mean .

SymPy: a(b(1)) = a(4) = 4

Knuth, Donald (1973). The art of computer programming. Reading, Mass: Addison-Wesley Pub. Co. ISBN 0201896850. — Chapter 7.2.1.2. Generating all permutations


Cameron 1994[edit]

Cameron decribes the active form of a permutation as a one-to-one mapping from to itself, and the passive form as the ordered n-tuple .

I wrote for the result of applying the function to the element .
[...] In order that the result of applying first and then can be called , it is more natural to denote the image of under as .
[Footnote:] We say that permutations act on the right if they compose according to this rule.

What Cameron calls the passive permutation is simply the one-line or word representation of an (active) permutation, not a passive permutation in the sense of this article.

Cameron, Peter J. (1994), Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, ISBN 0-521-45761-0  — Chapter 3.5 Permutations (p. 29)

Grimaldi 2004[edit]

Illustrations of and as in the book, illustration of added

Grimaldi defines as the left rotation of a triangle and as the reflection that leaves the right base angle in place.
Their product is the reflection about the vertical axis.

Grimaldi, Ralph (2004). Discrete and combinatorial mathematics : an applied introduction. Reading, Mass: Addison-Wesley Longman. ISBN 0321211030. — Example 16.7 (p.749)


References[edit]

  1. Some authors – see Cameron (1994) – call a permutations one-line notation the passive form. But this article calls a permutation passive when its one-line notation denotes where the arrows in its arrow diagram come from — as opposed to where they go. In the 19th century passive permutations were called permutations, while permutations in the modern sense were called substitutions.
  2. Compare this list of the first 40320 finite permutations: https://oeis.org/A198380/a198380_1.txt (1-based)