Coerce a permutation to a function

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, if \(\alpha\colon X\times G\longrightarrow X\) is a function, it is a group action if \(\alpha(x,e)=e\) and \(\alpha(\alpha(x,h),h)=\alpha(x,gh)\). Writing \(x\cdot g\) for \(\alpha(x,g)\) we have \(x\cdot e=x\) and \((x\cdot g)\cdot h=x\cdot(gh)\). In package idiom, given permutation g [considered as an element of the symmetric group \(S_n\)], we return the function with domain \(\left[n\right]=\left\lbrace 1,\ldots,n\right\rbrace\) mapping \(x\in[n]\) to \(\alpha(g,x)\). For example, if \(g=(172)(45)\) then \(\alpha(g,7)=2\) and \(\alpha(g,4)=5\). 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
  

Author

Robin K. S. Hankin

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.

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)
#>  [1] 7 7 9 1 7 5 6 3 8 4
as.function(as.cycle(1:9))(sample(9))
#> [1] 2 7 3 8 6 1 9 4 5
as.function(allcyc(5:8))(1:6)
#> [1] 1 2 3 4 8 5