VerticalSeparators

Interior positions i with |p(i-1) - p(i+1)| = 1.

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VerticalSeparators(p)Interior positions i with |p(i-1) - p(i+1)| = 1.

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) < 3 else Count(Filter(
                                       Range(2, Length(_x) - 1),
                                       Function(
                                                 Abs(_x[i - 1] - _x[i + 1]) == 1,
                                                 i,
                                               ),
                                     ))

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001403

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001403

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001403

About

Examples	6
Crosswalk	FindStat