OccurrencesOf3241Or4231

Quadruples ordered as 3241 or as 4231.

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OccurrencesOf3241Or4231(p)Quadruples ordered as 3241 or as 4231.

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.
• A permutation avoids both exactly when it is an input-restricted deque.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 4 else Sum(Map(Function(Sum(Map(
                                                   Function(Sum(Map(
                                                                     Function(Count(Filter(
                                                                                            Range(k + 1, Length(_x)),
                                                                                            Function(_x[l] < _x[j] &&
                                                                                                     _x[j] < _x[i] &&
                                                                                                     _x[i] < _x[k] ||
                                                                                                     _x[l] < _x[j] &&
                                                                                                     _x[j] < _x[k] &&
                                                                                                     _x[k] < _x[i], l),
                                                                                          )), k),
                                                                     Range(
                                                                            j +
                                                                            1,
                                                                            Length(_x) -
                                                                            1,
                                                                          ),
                                                                   )), j),
                                                   Range(i + 1, Length(_x) - 2),
                                                 )), i), Range(1, Length(_x) - 3)))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{OccurrencesOf3241Or4231}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOf3241Or4231}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{OccurrencesOf3241Or4231}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{OccurrencesOf3241Or4231})$Out$0$

Known value · FindStat St000404

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{OccurrencesOf3241Or4231})$Out$0$

Known value · FindStat St000404

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{OccurrencesOf3241Or4231})$Out$0$

Known value · FindStat St000404

About

Examples	6
Crosswalk	FindStat