The hardware and bandwidth for this mirror is donated by dogado GmbH, the Webhosting and Full Service-Cloud Provider. Check out our Wordpress Tutorial.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]dogado.de.

Function as.function.permutation() in the permutations package: group actions

Robin K. S. Hankin

as.function.permutation

## function (x, ...) 
## {
##     a <- NULL
##     x <- as.matrix(as.word(x))
##     if (nrow(x) == 1) {
##         return(as.function(alist(a = , x[, a])))
##     }
##     else {
##         return(as.function(alist(a = , x[cbind(seq_len(nrow(x)), 
##             a)])))
##     }
## }
## <bytecode: 0x64414f1c07b0>
## <environment: namespace:permutations>

To cite the permutations package in publications, please use Hankin (2020). The permutations package was intended to manipulate and combine permutations, but often one wants to consider the effect of a permutation on the underlying set, taken to be \(\left[n\right]=\left\lbrace 1,2,\ldots,n\right\rbrace\). In other words, we wish to consider a permutation as a function. In package idiom, coercing a permutation to a function is straightforward:

g <- as.cycle("(45)(127)")
as.function(g)(4)

## [1] 5

Above we see that permutation \((45)(127)\) maps 4 to 5. We can see from the function body, at the top of the page, that permutations are coerced to word form. Function as.function.permutation() uses as.matrix() to stop “x[a,]” dispatching to [.word() and use matrix extraction instead. It might be argued that unclass() would be better coding.

Coercion is vectorized:

as.function(g)(1:7)

## [1] 2 7 3 5 4 6 1

as.function(allperms(4))(3)

##  [1] 3 4 2 4 2 3 3 4 1 4 1 3 2 4 1 4 1 2 2 3 1 3 1 2

as.function(rperm(7,8))(1:7)

## [1] 6 2 2 7 3 2 2

The second and third forms use the alist(a = , x[cbind(seq_len(nrow(x)),a)]) construction. We now discuss the extent to which the underlying permutation group is represented in package idiom. Consider the following construction:

(p <- cyc_len(2))

## [1] (12)

as.function(p)(3)

## Error in x[, a]: subscript out of bounds

On the one hand, object p is a permutation on the set \([2]=\left\lbrace 1,2\right\rbrace\). The action of this permutation on 3 is not defined, and the package returns an error. Above we effectively see

t(1:2)[,3]

## Error in t(1:2)[, 3]: subscript out of bounds

which is the origin of the error. On the other hand, one might reasonably hold that the action of \((12)\) on 3 should be 3, on the grounds that \((12)\) transposes elements 1 and 2 and leaves all other elements unchanged. To realise this interpretation we need to ensure that p has underlying set including 3, in this case \(\left\lbrace 1,2,3\right\rbrace\). This is straightforward with as.word():

as.function(as.word(p,n=3))(3)

## [1] 3

as.function(id)(4)

## Error in x[, a]: subscript out of bounds

as.function(as.word(id,n=4))(4)

## [1] 4

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.