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] (26)(34) (195)    (38764)  (59768)  (16875)  (28)(67) (679)    (28476) 
z[is.odd(z)]
#>  [1] (2659)    (1946)    (379)(45) (164)(35) (58)      (28)(367) (146)(37)
#>  [8] (3897)    (18)      (29)      (1687)    (45)