Mad

Foata and Zeilberger's mad: 2 (2-31) + (31-2) + (21).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000795via Permutation✓↗

Mad(p)Foata and Zeilberger's mad: 2 (2-31) + (31-2) + (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.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000795

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000795

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

live ↗

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

Known value · FindStat St000795

About

Examples	6
Crosswalk	FindStat