permutations (version 1.1-5)

sgn: Sign of a permutation

Description

Returns the sign of a permutation

Usage

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

Arguments

x

permutation object

Author

Robin K. S. Hankin

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.

See Also

shape

Examples

sgn(id)  # always problematic

sgn(rperm(10,5))

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)]
z[is.odd(z)]