Alignments

Pairs i, j of arcs of the permutation diagram that align.

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Alignments(p)Pairs i, j of arcs of the permutation diagram that align.

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 1 else Sum(Map(
                                  Function(
                                            Count(Filter(
                                                          Range(1, Length(_x)),
                                                          Function(j < i && i <
                                                                            _x[i] &&
                                                                   _x[i] < _x[j] || _x[j] <
                                                                       _x[i] && _x[i] <=
                                                                           i && i <
                                                                               j ||
                                                                   j <= _x[j] &&
                                                                   _x[j] < _x[i] && _x[i] < i ||
                                                                   _x[i] < i &&
                                                                   i < j && j <=
                                                                       _x[j], j),
                                                        )),
                                            i,
                                          ),
                                  Range(1, Length(_x)),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{Alignments}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{Alignments}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$1$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{Alignments})$Out$0$

Known value · FindStat St000222

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{Alignments})$Out$0$

Known value · FindStat St000222

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{Alignments})$Out$0$

Known value · FindStat St000222

About

Examples	5
Crosswalk	FindStat