WeakMajorIndex

The sum of the positions i with c(i) >= c(i+1).

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001930via BinaryWord✓↗St001931via IntegerComposition✓↗

WeakMajorIndex(composition)The sum of the positions i with c(i) >= c(i+1).

Typed signatures

Type	Package	Applies
(binary_word | composition | list<integer>) -> number	combinatorics	always

Domain: Composition statistics

Details

• Defined over IntegerComposition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerComposition, 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.
• Reads the composition as a word, so it stands on a bare list too.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

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

on BinaryWord, over _x

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

Examples

Basic #

the composition $[2,1,3,1,1]$

live ↗

In$\mathrm{WeakMajorIndex}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$8$

the composition $[2,1,3,1,1]$, as a plain list

live ↗

In$\mathrm{WeakMajorIndex}(\bigl\lbrack2, 1, 3, 1, 1\bigr\rbrack)$Out$8$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St001931

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([2]), \text{WeakMajorIndex})$Out$0$

Known value · FindStat St001931

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{WeakMajorIndex})$Out$0$

Known value · FindStat St001931

About

Examples	5
Crosswalk	FindStat ×2