Function `swap()` function in the `permutations` package

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

set.seed(0)
library("permutations")
options(rmarkdown.html_vignette.check_title = FALSE)
knitr::opts_chunk$set(echo = TRUE)
knit_print.function <- function(x, ...){dput(x)}
registerS3method(
  "knit_print", "function", knit_print.function,
  envir = asNamespace("knitr")
)

knitr::include_graphics(system.file("help/figures/permutations.png", package = "permutations"))

swap

To cite the permutations package in publications, please use @hankin2020. Function swap() returns length-two cycles corresponding to swapping its two arguments:

swap(3, 6)

If the two arguments are identical, the identity is returned:

swap(7, 7)

It is vectorized:

swap(1, 1:8)
swap(1:8, 8:1)

Sometimes, printing in word form reveals structure:

print_word(swap(9:1, 1:9))

We may use swap() to effect substitution:

options(perm_set = letters)
options(comma = FALSE)
allcyc(1:4)
allcyc(1:4) ^ swap(4, 26)  # replace "d" with "z"

Vignette outer_automorphisms_of_S6 includes a nice use-case of swap(). Specifically, it uses the identity

$$ (abcd) = (1a)(1b)(1c)(1d)(1a),\qquad 1\not\in\left\lbrace b,c,d\right\rbrace$$

We note that if $a=1$ the identity reduces to $(1bcd) = (1b)(1c)(1d)$. However, if (say) $c=1$ the conditions of the inequality are not satisfied. The right hand side becomes

$$(1a)(1b)(1d)(1a)=(abd)\neq(ab1d)$$.

In package idiom this does not arise because if 1 is present in a cycle, it appears in the first place following nicify_cyclist(). If, for some reason, we cannot rely on the placing of a possible 1 we might use

$$(abcd) = (na)(nb)(nc)(nd)(na),\qquad n\not\in\left\lbrace a,b,c,d\right\rbrace.$$

Observe that $a,b,c,d$ must be distinct, for otherwise the left hand side is not well-defined. In the package:

as.cycle(c(3,4,3,6))

References

permutations documentation built on July 13, 2026, 5:07 p.m.