BruhatLowerCovers

Pairs i < j with p(i) > p(j) and no entry between them valued between p(j) and p(i).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000957via Permutation✓↗

BruhatLowerCovers(p)Pairs i < j with p(i) > p(j) and no entry between them valued between p(j) and p(i).

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.
• The boxed occurrences of 21.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000957

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000957

FindStat's value for [1,2,3]

live ↗

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

Known value · FindStat St000957

About

Examples	6
Crosswalk	FindStat