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.

1 2 3 4 5 | ```
standard(cyc,n=NULL)
standard_cyclist(x,n=NULL)
fbin_single(vec)
fbin(W)
fbin_inv(cyc)
``` |

`vec` |
In function |

`W` |
In functions |

`cyc` |
In functions |

`n` |
In function |

`x` |
In function |

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).

Robin K. S. Hankin

R. P. Stanley 2011 *Enumerative Combinatorics*

1 2 3 4 5 6 7 8 9 10 11 12 |

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