fbin

0th

Percentile

The fundamental bijection

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 recoverd by inserting a left parenthesis preceding every left-to-right maximum, and right parentheses where appropriate.

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

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

nicify_cyclist

Aliases
• fbin
• fbin_single
• fbin_inv
• standard
• standard_cyclist
Examples
# NOT RUN {
# Stanley's example w:
standard(cycle(list(list(c(1,4),c(3,7,5)))))

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))

# }

Documentation reproduced from package permutations, version 1.0-5, License: GPL-2

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