FirstDescent

The smallest descent position, or n when p is increasing.

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FirstDescent(p)The smallest descent position, or $n$ when $p$ is increasing.

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 St000654: p(n+1) = 0, so position n always descends.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

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

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St000654

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St000654

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

live ↗

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

Known value · FindStat St000654

About

Examples	6
Crosswalk	FindStat

See also: FirstAscent, Descents