NonOverlappingDescents

Descents taken greedily left to right, none starting where the last one ended.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001928via Permutation✓↗

NonOverlappingDescents(p)Descents taken greedily left to right, none starting where the last one ended.

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.
• Each maximal descending run of length m contributes floor(m / 2).

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 2 else Fold(Function(
                                        [a[1] + 1, b] if _x[b] > _x[b + 1] && a[2] <
                                                                              b -
                                                                              1 else a,
                                        a,
                                        b,
                                      ), [0, -1], Range(1, Length(_x) - 1, 1))[1]

Examples

Basic #

the one-line word $312$

live ↗

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

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

live ↗

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

the one-line word $2413$

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001928

FindStat's value for [1,2]

live ↗

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

Known value · FindStat St001928

FindStat's value for [2,1]

live ↗

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

Known value · FindStat St001928

About

Examples	6
Crosswalk	FindStat