NonNegativeWalkSteps

Steps of the walk (up at an ascent, down at a descent) that begin and end at height at least 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001332via Permutation✓↗

NonNegativeWalkSteps(p)Steps of the walk (up at an ascent, down at a descent) that begin and end at height at least 0.

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 Count(Filter(
                                       Range(1, Length(_x) - 1),
                                       Function(2 * Count(Filter(
                                                                  Range(1, i, 1),
                                                                  Function(_x[j] <
                                                                           _x[j +
                                                                              1], j),
                                                                )) - i >=
                                                (1 if _x[i] < _x[i + 1] else 0), i),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001332

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001332

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001332

About

Examples	6
Crosswalk	FindStat