NonRecords

Positions that are none of a left-to-right maximum or minimum, or a right-to-left maximum or minimum.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001715via Permutation✓↗

NonRecords(p)Positions that are none of a left-to-right maximum or minimum, or a right-to-left maximum or minimum.

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(1, i),
                                                                           )) && _x[i] <
                                                                        Max(Map(
                                                                                 Function(_x[j], j),
                                                                                 Range(i, Length(_x)),
                                                                               )) && _x[i] >
                                                                            Min(Map(
                                                                                     Function(_x[j], j),
                                                                                     Range(i, Length(_x)),
                                                                                   )), i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

In$\mathrm{NonRecords}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{NonRecords}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001715

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001715

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001715

About

Examples	6
Crosswalk	FindStat