WeakRecordCount

Parts at least as great as every part before.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000765via IntegerComposition✓↗

WeakRecordCount(composition)Parts at least as great as every part before.

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 Count(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$\mathrm{WeakRecordCount}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$2$

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000765

FindStat's value for [2]

live ↗

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

Known value · FindStat St000765

FindStat's value for [3]

live ↗

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

Known value · FindStat St000765

About

Examples	4
Crosswalk	FindStat