permutations (version 1.1-5)

fbin: The fundamental bijection

Description

Stanley defines the fundamental bijection on page 30.

Given \(w=(14)(2)(375)(6)\), Stanley writes it in standard form (specifically: each cycle is written with its largest element first; cycles are written in increasing order of their largest element). Thus we obtain \((2)(41)(6)(753)\).

Then we obtain \(w^*\) from \(w\) by writing it in standard form an erasing the parentheses (that is, viewing the numbers as a word); here \(w^*=2416753\).

Given this, \(w\) may be recovered by inserting a left parenthesis preceding every left-to-right maximum, and right parentheses where appropriate.

Function standard() takes an object of class cycle and returns a list of cyclists. NB this does not return an object of class “cycle” because cycle() calls nicify().

Function standard_cyclist() retains length-one cycles (compare nicify_cyclist(), which does not).

Usage

standard(cyc,n=NULL)
standard_cyclist(x,n=NULL)
fbin_single(vec)
fbin(W)
fbin_inv(cyc)

Arguments

vec

In function fbin_single(), an integer vector

W

In functions fbin() and fbin_inv(), an object of class permutation, coerced to word and cycle form respectively

cyc

In functions fbin_single() and standard(), permutation object coerced to cycle form

n

In function standard() and standard_cyclist(), size of the partition to assume, with default NULL meaning to use the largest element of any cycle

x

In function standard_cyclist(), a cyclist

Author

Robin K. S. Hankin

Details

The user-friendly functions are fbin() and fbin_inv() which perform Stanley's “fundamental bijection”. Function fbin() takes a word object and returns a cycle; function fbin_inv() takes a cycle and returns a word.

The other functions are low-level helper functions that are not really intended for the user (except possibly standard(), which puts a cycle object in standard order in list form).

References

R. P. Stanley 2011 Enumerative Combinatorics

See Also

nicify_cyclist

Examples

# Stanley's example w:
standard(cycle(list(list(c(1,4),c(3,7,5)))))

standard_cyclist(list(c(4, 6), c(7), c(2, 5, 1), c(8, 3)))


w_hat <- c(2,4,1,6,7,5,3)

fbin(w_hat)
fbin_inv(fbin(w_hat))


x <- rperm(40,9)
stopifnot(all(fbin(fbin_inv(x))==x))
stopifnot(all(fbin_inv(fbin(x))==x))