LeftToRightMaximaXorRightToLeftMinima

Positions that are a left-to-right maximum or a right-to-left minimum but not both.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001005via Permutation✓↗

LeftToRightMaximaXorRightToLeftMinima(p)Positions that are a left-to-right maximum or a right-to-left minimum but not both.

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) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] == Max(Map(
                                                                  Function(
                                                                            _x[j],
                                                                            j,
                                                                          ),
                                                                  Range(1, i),
                                                                )) && _x[i] >
                                                                     Min(Map(
                                                                              Function(_x[j], j),
                                                                              Range(i, Length(_x)),
                                                                            )) || _x[i] ==
                                                                         Min(Map(
                                                                                  Function(_x[j], j),
                                                                                  Range(i, Length(_x)),
                                                                                )) && _x[i] <
                                                                             Max(Map(
                                                                                      Function(_x[j], j),
                                                                                      Range(1, i),
                                                                                    )), i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

In$\mathrm{LeftToRightMaximaXorRightToLeftMinima}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$3$

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001005

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001005

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

live ↗

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

Known value · FindStat St001005

About

Examples	6
Crosswalk	FindStat