WeakRecordPositionSum

The sum of the positions of the weak records.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000762via IntegerComposition✓↗

WeakRecordPositionSum(composition)The sum of the positions of the weak records.

Typed signatures

Type	Package	Applies
(composition) -> 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 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.
• Knopfmacher and Mansour, Record statistics in integer compositions.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

0 if Length(_x) < 1 else Sum(Filter(
                                     Range(1, Length(_x)),
                                     Function(
                                               Count(Filter(
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                             Function(
                                                                       j < i &&
                                                                       _x[j] >
                                                                       _x[i],
                                                                       j,
                                                                     ),
                                                           )) == 0,
                                               i,
                                             ),
                                   ))

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [2]

live ↗

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

Known value · FindStat St000762

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{WeakRecordPositionSum})$Out$1$

Known value · FindStat St000762

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([4]), \text{WeakRecordPositionSum})$Out$1$

Known value · FindStat St000762

About

Examples	4
Crosswalk	FindStat