PathDepth

The most zeros less ones among the prefixes.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000877via BinaryWord✓↗

PathDepth(word)The most zeros less ones among the prefixes.

Typed signatures

Type	Package	Applies
(binary_word) -> number	combinatorics	always

Domain: Binary word statistics

Details

• Defined over BinaryWord as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a BinaryWord 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.
• Reads the word as a path with a 0 a step up and a 1 a step down; the empty prefix counts, so the depth is at least 0.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

Fold(Function([a[1] + (1 - 2b), Max(a[2], a[1] + (1 - 2b))], a, b), [0, 0], _x)[2]

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{PathDepth}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$0$

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St000877

FindStat's value for 1

live ↗

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

Known value · FindStat St000877

FindStat's value for 00

live ↗

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

Known value · FindStat St000877

About

Examples	4
Crosswalk	FindStat