ExteriorLength

The longest proper prefix whose relative order is that of the suffix of the same length.

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ExteriorLength(p)The longest proper prefix whose relative order is that of the suffix of the same length.

Typed signatures

Type	Package	Applies
(list<integer> | 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 also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 2 else Max(Map(
                                  Function(Length(_x) - i, i),
                                  Filter(
                                          Range(1, Length(_x) - 1),
                                          Function(Max(Sum(Map(
                                                                Function(Count(Filter(
                                                                                       Range(1, Length(_x) - i, 1),
                                                                                       Function(b > a &&
                                                                                                (-1 if _x[a] < _x[b] else 1 if _x[a] > _x[b] else 0) !=
                                                                                                (-1 if _x[a + i] <
                                                                                                       _x[b + i] else 1 if _x[a + i] >
                                                                                                                  _x[b + i] else 0), b),
                                                                                     )), a),
                                                                Range(
                                                                       1,
                                                                       Length(_x) -
                                                                       i,
                                                                       1,
                                                                     ),
                                                              )), 0) == 0, i),
                                        ),
                                ))

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $312$, as a plain list

live ↗

In$\mathrm{ExteriorLength}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$1$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001052

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001052

FindStat's value for [1,2,3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2, 3]), \text{ExteriorLength})$Out$2$

Known value · FindStat St001052

About

Examples	6
Crosswalk	FindStat