BabsonSteingrimssonStatisticDoublePrime

Descents plus the occurrences of the vincular patterns 1-32, 3-12 and 3-21.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000797via Permutation✓↗

BabsonSteingrimssonStatisticDoublePrime(p)Descents plus the occurrences of the vincular patterns 1-32, 3-12 and 3-21.

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.
• FindStat's stat'', after Babson and Steingrimsson.

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000797

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000797

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

live ↗

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

Known value · FindStat St000797

About

Examples	6
Crosswalk	FindStat