# permutations v1.0-9

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## The Symmetric Group: Permutations of a Finite Set

Manipulates invertible functions from a finite set to itself. Can transform from word form to cycle form and back.

# Overview

The permutations package provides R-centric functionality for working with permutations of a finite set. It includes group-theoretic composition of permutations and can transform from word form to cycle form and back.

# Installation

You can install the released version of permutations from CRAN with:

# install.packages("permutations")  # uncomment this to use the package
library("permutations")
#>
#> Attaching package: 'permutations'
#> The following object is masked from 'package:stats':
#>
#>     cycle


The package is maintained on github.

# The permutations package in use

Random permutations on a finite set are given by the rperm() command:

rperm(10,9)
#>  [1] (19524)(37)    (1238794)      (1745682)(39)  (15)(3897)(46) (132654789)
#>  [6] (17263)(59)    (136)(47589)   (162)(4795)    (14763)(259)   (168347925)


The result is printed in cycle form but we can print in word form if we wish:

options(print_word_as_cycle=FALSE)  # override default
as.word(rperm(10,9,3))
#>      {1} {2} {3} {4} {5} {6} {7} {8} {9}
#> [1]  7   .   .   .   .   .   9   .   1
#> [2]  .   9   2   .   .   .   .   .   3
#> [3]  .   .   .   .   .   .   .   .   .
#> [4]  3   .   1   .   .   .   .   .   .
#> [5]  .   8   .   .   2   .   .   5   .
#> [6]  .   .   .   6   .   4   .   .   .
#> [7]  .   .   .   .   .   .   .   .   .
#> [8]  .   .   .   8   .   .   .   4   .
#> [9]  6   .   1   .   .   3   .   .   .
#> [10] .   .   .   9   .   .   .   .   4
options(print_word_as_cycle=TRUE)  # usually want to print a cycle irregardless


(A dot indicates a fixed point). The package uses arithmetic operations * to combine permutations and ^ for conjugation:

(a <- as.word(c(4,2,3,1,5,7,6)))
#> [1] (14)(67)
(b <- as.cycle(1:4))
#> [1] (1234)
a*b
#> [1] (234)(67)
b*a
#> [1] (123)(67)


# The megaminx

The megaminx is a dodecahedral puzzle with similar construction to the Rubik cube puzzle that has 50 movable pieces and 132 coloured stickers (facets’’). The permutations package includes functionality to simulate the megaminx and exhibits an 82-turn superflip. The vignette gives an extended discussion.

## Functions in permutations

 Name Description orbit Orbits of integers permorder The order of a permutation Ops.permutation Arithmetic Ops Group Methods for permutations allperms All permutations of a given size cayley Cayley tables for permutation groups conjugate Are two permutations conjugate? cyclist details of cyclists fixed Fixed elements fbin The fundamental bijection as.function.permutation Coerce a permutation to a function c Concatenation of permutations rperm Random permutations print Print methods for permutation objects commutator Group-theoretic commutator and group action megaminx megaminx derangement Tests for a permutation being a derangement inverse Inverse of a permutation length Various vector-like utilities for permutation objects. get1 Retrieve particular cycles or components of cycles id The identity permutation dodecahedron The dodecahedron group permutation Functions to create and coerce word objects and cycle objects shape Shape of a permutation sgn Sign of a permutation permutations-package permutations megaminx_plotter Plotting routine for megaminx sequences tidy Utilities to neaten permutation objects size Gets or sets the size of a permutation valid Functions to validate permutations nullperm Null permutations perm_matrix Permutation matrices No Results!