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Permutation groups extension for Numbas

This extension adds a new data type to the Numbas JME system, representing elements of permutation groups (really, just S_infinity, but you can pretend you're working in an S_n).

Permutations are stored as bijections on the natural numbers.

The symbol for the identity permutation is e by default. You can change this by setting Numbas.extensions.permutations.identity_symbol in the question preamble.

JME data type

This extension adds a new JME data type, Numbas.jme.types.permutation, representing a permutation.

JME Functions

permutation(map) or perm(map)

Create a permutation. map is a list of numbers. map[i] is the image of i under the permutation.

Note: Because lists are zero-indexed, the first element is 0, not 1.

Example: permutation([1,0]) is the permutation swapping the first two elements and will be displayed in cycle notation as (1,2).

permutation(str) or perm(str)

Create a permutation. str is a permutation in disjoint cycle notation.

Example: permutation("(1,2)(3,4,5)")

cycle(list)

Create a permutation consisting of a single cycle, whose elements are the given list. The elements are zero-indexed, like for permutation.

Example: cycle([0,4,1]) = permutation([4,0,2,3,1])

transposition(a,b)

A permutation representing the transposition of the elements a and b.

rotation(n,r)

Create a permutation representing a rotation of the numbers 1 to n by r places. If r is not given it defaults to 1.

flip(n)

Create a permutation representing a reflection of the numbers 1 to n.

compose(p1,p2) or p1*p2

Compose permutations. The action is on the left - (p1*p2)[x] = p1[p2[x]].

p^n

The nth power of p. p^0 is the identity permutation.

p[x]

Apply permutation p to the number x. Consider p as an element of S_n; any x greater than or equal to n is mapped to itself.

inverse(p)

Returns the inverse of permutation p.

even(p)

Is p an even permutation (can it be written as a product of an even number of transpositions?)

size(p)

The largest number n that is permuted by p, or equivalently, the smallest n such that p is a member of S_n.

order(p)

The order of p: the smallest power of p equivalent to the identity permutation.

cycles(p)

Calculate all of p's cycles. Returns a list of lists of numbers.

nontrivial_cycles(p)

Calculate all of p's cycles of length greater than 1.

show(p)

Render p as a TeX string (just substituting p into a TeX expression will also work, but you can use this if you want to manipulate the TeX string somehow)

twoline(p)

Render p in two-line form (numbers 1..n on top row, their images on the bottom) as a TeX string

as_transpositions(p)

Render p as a product of transpositions, in TeX.

p1=p2

Are p1 and p2 the same permutation? True if p1*inverse(p2) maps n to n for all n.

is_disjoint(str)

Returns true if str is a representation of a permutation as disjoint cycles (no two cycles have an element in common).

is_transpositions(str)

Returns true if str is a representation of a permutation as transpositions (each cycle has length 2).

is_rotation(p)

Returns true if p is a rotation, i.e. a permutation of the form f: i -> (i+x) mod N, for some 0< x < N.

is_flip(p)

Returns true if p is a flipped rotation, i.e. a permutation of the form f: i -> (x-i) mod N, for some 0<x<N.

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