Permutation notation
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.
The usual way is as an active permutation or map or substitution:
 moves an object from place to place . In the arrow diagram the oneline 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 oneline 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 isR(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  edit source]
Active[edit  edit source]
If permuting by gives the permutation is active.
Active left[edit  edit source]
AL 

If the notation should be seen as active left. ^{*} means and can be written . can be written . Permuting by may be written in different ways: The notation with the dot is just an abbreviation one may define. With two permutations it means: 
Active right[edit  edit source]
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 
Passive[edit  edit source]
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 5element permutations[edit  edit source]
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 1based 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.)
Box 0 Initialisation for the computations 

Python: The simple function from sympy.combinatorics import Permutation def before(positions, sequence): return [sequence[i] for i in positions] def inv(perm): length = len(perm) if sorted(perm) != list(range(length)): print('Inverse not possible!') else: inverse = [0] * length for key, val in enumerate(perm): inverse[val] = key return inverse p = [1, 3, 0, 2, 4] q = [4, 3, 2, 1, 0] r = before(p, q) P = Permutation(p) Q = Permutation(q) R = P * Q v = ['0', '1', '2', '3', '4'] e = [0, 0, 4, 2, 3, 2] Wolfram: p = {2,4,1,3,5} q = {5,4,3,2,1} v = {v1,v2,v3,v4,v5} e = {1,1,5,3,4,3} 
Active vs. passive[edit  edit source]
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).
Box 1 Ambiguity of  

Passive (PL):  Active (AR):  
If it is only required, that , and no one intends to permute vectors, there is no need to choose an interpretation. If on the other hand it is only required that a permutation permutes the vector into , it does dot matter if it is labelled or . While the permutations on the left and right have the same labels and do different things, the permutations below do exactly the same as those above them, but are labelled differently.

Left vs. right[edit  edit source]
Box 2 Ambiguity of with active permutations  

Left (AL):  Right (AR):  
Box 3 Ambiguity of with passive permutations  

Left (PL):  Right (PR):  
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  edit source]
Box 4 Formulas for and in twoline and cycle notation  

AL:  AR: 




The top left row is reordered like the bottom right, so the bottom left row becomes the result.  The top right row is reordered like the bottom left, so the bottom right row becomes the result. 
Box 5 Computation (AR only)  


Box 6  


Box 7  


Box 8  


Passive[edit  edit source]
Box 9  


Box 10  


Box 11  


Vectors, scalars and nonbijective maps[edit  edit source]
Active[edit  edit source]
Box 12 Permuting a vector  


Box 13 Concatenation with a scalar as permuting a vector  


Box 14 Concatenation with a scalar as concatenation with a constant  


Box 15 Concatenation with a nonbijective map  


Passive[edit  edit source]
Box 16 Permuting a vector  


Box 17 Concatenation with a scalar as permuting a vector  


Examples with permutations of the square[edit  edit source]
The dihedral group Dih_{4} 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  edit source]
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.
permutohedron section, cycle graph, Cayley table  

Active left (linear maps)[edit  edit source]
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  

Cayley tables  

The rotations of the square form the cyclic group C_{4}. The 2×2 matrices of the linear maps are a subgroup of SL(2,3), and within that of the quaternion group.  

Active vs. passive[edit  edit source]
Box 19 Ambiguity of  

Passive (PL):  Active (AR):  
The inverse of is , while and are selfconjugate. The Cayley table above contains , so it fits the row above, and it contains , so its transpose would fit the row below. As above (box 1) the permutations on the left and right have the same labels and do different things, while the permutations below do exactly the same as those above them, but are labelled differently.

Left vs. right[edit  edit source]
Active[edit  edit source]
Cayley tables  

Box 20  


Box 21  


Passive[edit  edit source]
Box 22  


Box 23  


Languages[edit  edit source]
Wolfram[edit  edit source]
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 1based 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 oneline 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)
More examples  

I: p O: {2,4,1,3,5} I: q O: {5,4,3,2,1} I: PermutationProduct[p, q] (* R *) O: {4,2,5,3,1} I: PermutationProduct[q, p] (* S *) O: {5,3,1,4,2} For two permutations I: PermutationReplace[p, q] (* R *) O: {4,2,5,3,1} I: PermutationReplace[q, p] (* S *) O: {5,3,1,4,2}
I: Permute[p, q] O: {5,3,1,4,2} I: Permute[q, p] O: {3,5,2,4,1}
Confusingly the result of The result of 
SymPy[edit  edit source]
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.
Negative results for ~p(i)


Inverse permutations behave strangely when accessed from the right.

Sources[edit  edit source]
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  edit source]
Aigner 2007[edit  edit source]
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  edit source]
Knuth 1973[edit  edit source]
In TAoCP Knuth states that in his book permutations are always multiplied from left to right, and gives the following example:
SymPy 

>>> from sympy.combinatorics import Permutation >>> a = Permutation(0, 2)(1, 5, 3) >>> b = Permutation(0, 5)(1, 4)(2, 3) >>> a * b Permutation(0, 3, 4, 1)(2, 5) >>> b * a Permutation(0, 3)(1, 4, 5, 2) >>> Permutation.print_cyclic = False >>> a Permutation[2, 5, 0, 1, 4, 3] >>> b Permutation[5, 4, 3, 2, 1, 0] >>> a * b Permutation[3, 0, 5, 4, 1, 2] >>> b * a Permutation[3, 4, 1, 0, 5, 2] 
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: AddisonWesley Pub. Co. ISBN 0201896850. — Chapter 7.2.1.2. Generating all permutations
Cameron 1994[edit  edit source]
Cameron decribes the active form of a permutation as a onetoone mapping from to itself, and the passive form as the ordered ntuple .
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 oneline 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 0521457610 — Chapter 3.5 Permutations (p. 29)
Grimaldi 2004[edit  edit source]
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: AddisonWesley Longman. ISBN 0321211030. — Example 16.7 (p.749)
References[edit  edit source]
 ↑ Some authors – see Cameron (1994) – call a permutations oneline notation the passive form. But this article calls a permutation passive when its oneline notation denotes where the arrows in its arrow diagram come from — as opposed to where they go. In the 19^{th} century passive permutations were called permutations, while permutations in the modern sense were called substitutions.
 ↑ Compare this list of the first 40320 finite permutations: https://oeis.org/A198380/a198380_1.txt (1based)