OccurrencesOf321Or3412

Triples ordered as 321, plus quadruples ordered as 3412.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001411via Permutation✓↗

OccurrencesOf321Or3412(p)Triples ordered as 321, plus quadruples ordered as 3412.

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 boolean: its Bruhat interval below is a boolean lattice.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

(0 if Length(_x) < 3 else Sum(Map(Function(Sum(Map(
                                                    Function(Count(Filter(
                                                                           Range(j + 1, Length(_x)),
                                                                           Function(_x[k] < _x[j] &&
                                                                                    _x[j] < _x[i], k),
                                                                         )), j),
                                                    Range(i + 1, Length(_x) - 1),
                                                  )), i), Range(
                                                                1,
                                                                Length(_x) - 2,
                                                              )))) +
(0 if Length(_x) < 4 else Sum(Map(Function(Sum(Map(
                                                    Function(Sum(Map(
                                                                      Function(Count(Filter(
                                                                                             Range(k + 1, Length(_x)),
                                                                                             Function(_x[k] < _x[l] &&
                                                                                                      _x[l] < _x[i] &&
                                                                                                      _x[i] < _x[j], 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$\mathrm{OccurrencesOf321Or3412}({\operatorname{Permutation}}([3, 1, 2]))$Out$0$

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

live ↗

In$\mathrm{OccurrencesOf321Or3412}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$0$

the one-line word $2413$

live ↗

In$\mathrm{OccurrencesOf321Or3412}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$0$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{OccurrencesOf321Or3412})$Out$0$

Known value · FindStat St001411

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{OccurrencesOf321Or3412})$Out$0$

Known value · FindStat St001411

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{OccurrencesOf321Or3412})$Out$0$

Known value · FindStat St001411

About

Examples	6
Crosswalk	FindStat