permutations (version 1.0-9)

# 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.

## 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

## See Also

`nicify_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))

# }
```