LeftToRightMinima

Positions smaller than everything before them.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000542via Permutation✓↗

LeftToRightMinima(p)Positions smaller than everything before them.

SageMathenumeratio_stat("Permutation", p, "LeftToRightMinima")

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] == Min(Map(
                                                                  Function(
                                                                            _x[j],
                                                                            j,
                                                                          ),
                                                                  Range(1, i),
                                                                )), i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000542

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000542

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000542

About

Examples	6
Checked against	SageMath 6 agree
Crosswalk	FindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts