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

Robin K. S. Hankin

as.function.permutation

## function (x, ...) 
## {
##     a <- NULL
##     x <- unclass(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: 0x559ac91f05d0>
## <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

Permutations of sets other than integers

Although the most natural underlying set for a permutation is the integers, the print method can use different sets. One natural set to use is letters:

set.seed(0)
options(perm_set = letters)
P <- rperm()
P

##  [1] (afb)(cd)(eg) (agedf)(bc)   (ae)(cfd)     (aegdb)(cf)   (abfec)      
##  [6] (acbf)(de)    (adfcbge)     (bdcge)       (acfgd)       (afgdeb)     
## [coerced from word form]

What we want to do is have as.function(P)("a") return "f", "g", "e" etc. We may coerce P to a function, but this operates [for good reason] on the integers, not letters:

as.function(P)(1)

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

Above we see that as.function(P) returns integers, whose print method is not affected by option perm_set. We can get some of the desired functionality by using base R extraction:

letters[as.function(P)(1)]

##  [1] "f" "g" "e" "e" "b" "c" "d" "a" "c" "f"

but we would like to pass a named letter such as "b" (not a number) to the function. Currently, the only way to do it is somewhat klunky:

letters[as.function(P)(which(letters == "b"))]

##  [1] "a" "c" "b" "a" "f" "f" "g" "d" "b" "a"

Above we see that P[1]("b") = "a", P[2]("b") = "c", and so on. Vectorizing this functionality in the argument is even more klunky. Suppose we wish to determine P[1]("a"), P[2]("b"), etc:

(w <- letters[1:10])

##  [1] "a" "b" "c" "d" "e" "f" "g" "h" "i" "j"

P <- as.word(P, n=10)
letters[as.function(P)(sapply(w, \(x){which(x==letters)}))]

##  [1] "f" "c" "f" "b" "c" "a" "e" "h" "i" "j"

Above we see that P[1]("a") = "f", P[2]("b") = "c", etc.

Note on identity permutation

The ever-problematic identity permutation acts on the empty set so its functionalization always returns an error:

as.function(id)(4)

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

Again the resolution is to coerce to word form with explicit n:

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

## [1] 4

References

Hankin, R. K. S. 2020. “Introducing the Permutations R Package.” SoftwareX 11. https://doi.org/10.1016/j.softx.2020.100453.