cyclist: details of cyclists

Description Usage Arguments Details Author(s) See Also Examples

Description

Various functionality to deal with cyclists

Usage

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Arguments

p

Integer vector, interpreted as a word

x,cyc

A cyclist

n

In function cycle2word_single(), the size of the permutation to induce

rm1,smallest_first

In function nicify_cyclist(), Boolean, governing whether or not to remove length-1 cycles, and whether or not to place the smallest element in each cycle first (non-default values are used by standard_cyclist())

Details

A cyclist is an object corresponding to a permutation P. It is a list with elements that are integer vectors corresponding to the cycles of P. This object is informally known as a cyclist, but there is no S3 class corresponding to it.

An object of S3 class cycle is a (possibly named) list of cyclists. NB: there is an unavoidable notational clash here. When considering a single permutation, “cycle” means group-theoretic cycle; when considering R objects, “cycle” means “an R object of class cycle whose elements are permutations written in cycle form”.

The elements of a cyclist are the disjoint group-theoretic cycles. Note the redundancies inherent: firstly, because the cycles commute, their order is immaterial (and a list is ordered); and secondly, the cycles themselves are invariant under cyclic permutation. Heigh ho.

A cyclist may be poorly formed in a number of ways: the cycles may include repeats, or contain elements which are common to more than one cycle. Such problems are detected by cycle.valid(). Also, there are less serious problems: the cycles may include length-one cycles; the cycles may start with an element that is not the smallest. These issues are dealt with by nicify_cyclist().

Author(s)

Robin K. S. Hankin

See Also

as.cycle,fbin,valid

Examples

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vec2cyclist_single(c(7,9,3,5,8,6,1,4,2))

char2cyclist_single("(342)(19)")

nicify_cyclist(list(c(4, 6), c(7), c(2, 5, 1), c(8, 3)))
nicify_cyclist(list(c(4, 6), c(7), c(2, 5, 1), c(8, 3)),rm1=TRUE)


cyclist2word_single(list(c(1,4,3),c(7,8)))

permutations documentation built on Nov. 13, 2020, 1:14 a.m.