BalancedPrefixCount

The nonempty prefixes with as many 0s as 1s.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001355via BinaryWord✓↗

BalancedPrefixCount(word)The nonempty prefixes with as many 0s as 1s.

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.
• The returns of the word, read as a lattice path, to the main diagonal.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

Fold(Function(
               [a[1] + (2b - 1), a[2] + (1 if a[1] + (2b - 1) == 0 else 0)],
               a,
               b,
             ), [0, 0], _x)[2]

Examples

Basic #

the word $110100111$

live ↗

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

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St001355

FindStat's value for 1

live ↗

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

Known value · FindStat St001355

FindStat's value for 00

live ↗

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

Known value · FindStat St001355

About

Examples	4
Crosswalk	FindStat