LongestStaircase

The greatest l with parts at positions i_1 < ... < i_l such that part i_m is at least m.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000758via IntegerComposition✓↗

LongestStaircase(composition)The greatest l with parts at positions i_1 < ... < i_l such that part i_m is at least m.

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.
• Skandera, An Eulerian partner for inversions (Definition 3.1): taking each part that exceeds the staircase so far is optimal.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

0 if Length(_x) < 1 else Fold(Function(a + 1 if b > a else a, a, b), 0, _x)

Examples

Basic #

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

live ↗

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

Properties #

FindStat's value for [1]

live ↗

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

Known value · FindStat St000758

FindStat's value for [2]

live ↗

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

Known value · FindStat St000758

FindStat's value for [3]

live ↗

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

Known value · FindStat St000758

About

Examples	4
Crosswalk	FindStat