MaximumOfDescentsAndInverseDescents

The larger of the number of descents and the number of inverse descents.

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MaximumOfDescentsAndInverseDescents(p)The larger of the number of descents and the number of inverse descents.

Typed signatures

Type	Package	Applies
(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 nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001489

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001489

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001489

About

Examples	5
Crosswalk	FindStat