permutations (version 1.1-6)

as.function.permutation: 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 permutation
as.function(x, ...)

Arguments

x

permutation

...

further arguments (currently ignored)

Author

Robin K. S. Hankin

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
  

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)))