as.function.permutation: Coerce a permutation to a function

In permutations: The Symmetric Group: Permutations of a Finite Set

Coerce a permutation to a function

Description

Coerce a permutation to an executable function with domain \left\lbrace 1,\ldots,n\right\rbrace.

The resulting function is vectorised.

Usage

## S3 method for class 'permutation'
as.function(x, ...)

Arguments

x	permutation
...	further arguments (currently ignored)

Details

This functionality is sometimes known as group action. Formally, suppose we have a set X, a group G, and a function \alpha\colon X\times G\longrightarrow X. Then we say \alpha is a group action if \alpha(x,e)=e and \alpha(\alpha(x,g),h)=\alpha(x,gh). Writing x\cdot g for \alpha(x,g) we have x\cdot e=x and (x\cdot g)\cdot h=x\cdot(gh). The dot may be omitted giving us (xg)h=x(gh). If the group is a permutation group on X, then it is natural to choose \alpha(x,g)=g(x).

In package idiom, given permutation g [considered as an element of the symmetric group S_n], as.function(g) returns the function with domain \left[n\right]=\left\lbrace 1,\ldots,n\right\rbrace mapping x\in[n] to \alpha(g,x)=g(x)=xg. For example, if g=(172)(45) then \alpha(g,7)=g(7)=7g=2 and similarly \alpha(g,4)=5.

Package idiom allows one to explicitly coerce g to a function, or to use the overloaded caret:

    (g <- as.cycle("(172)(45)"))
    #> [1] (172)(45)
    as.function(g)(7)
    #> [1] 2
    7^g
    #> [1] 2
  

Note

Multiplication of permutations loses associativity when using functional notation; see examples.

Also, note that the coerced function will not take an argument greater than the size (qv) of the permutation.

Vignette vignettes/groupaction.Rmd discusses this issue.

Author(s)

Robin K. S. Hankin

Examples

x <- cyc_len(3)
y <- cyc_len(5)

xfun <- as.function(x)
yfun <- as.function(y)

stopifnot(xfun(yfun(2)) == as.function(y*x)(2)) # note transposition of x & y

# postfix notation has the very appealing result x(fg) == (xf)g
# Carets are good too, in that x^(fg) == (x^f)^g.

a <- rperm()
b <- rperm()
stopifnot(2^(a*b) == (2^a)^b)

# it's fully vectorized:
as.function(rperm(10,9))(1)
as.function(as.cycle(1:9))(sample(9))
as.function(allcyc(5:8))(1:6)

# standard recycling rules apply:
f <- as.function(allperms(3))
all(f(1:3) == f(c(1:3,1:3)))

permutations documentation built on April 3, 2025, 7:09 p.m.