Sign of a permutation

Returns the sign of a permutation

Usage

sgn(x)
is.even(x)
is.odd(x)

Arguments

x

permutation object

Details

The sign of a permutation is \(\pm 1\) depending on whether it is even or odd. A permutation is even if it can be written as a product of an even number of transpositions, and odd if it can be written as an odd number of transpositions. The map \(\operatorname{sgn}\colon S_n\longrightarrow\left\lbrace +1,-1\right\rbrace\) is a homomorphism; see examples.

Author

Robin K. S. Hankin

Note

Internally, function sgn() coerces to cycle form.

The sign of the null permutation is NULL.

See also

shape

Examples

sgn(id)  # always problematic
#> [1] 1

sgn(rperm(10,5))
#>  [1] -1  1 -1  1  1  1 -1  1  1  1

x <- rperm(40,6)
y <- rperm(40,6)


stopifnot(all(sgn(x*y) == sgn(x)*sgn(y)))   # sgn() is a homomorphism


z <- as.cycle(rperm(20,9,5))
z[is.even(z)]
#>  [1] (498)    (19)(34) (485)    (248)    (35687)  (13264)  (13469)  (287)   
#>  [9] (19375)  (19)(57)
z[is.odd(z)]
#>  [1] (1352)    (37)(568) (127)(46) (1678)    (1392)    (15)(347) (1684)   
#>  [8] (2697)    (4879)    (1253)