LongestYamanouchiPrefix

The length of the longest prefix in which no prefix has more ones than zeros.

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LongestYamanouchiPrefix(word)The length of the longest prefix in which no prefix has more ones than zeros.

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.
• FindStat's St001371.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 1 else Min(Join(
                                   [Length(_x) + 1],
                                   Filter(
                                           Range(1, Length(_x)),
                                           Function(
                                                     2 * Max(Sum(Map(
                                                                      Function(_x[j], j),
                                                                      Range(
                                                                             1,
                                                                             1 +
                                                                             j -
                                                                             1,
                                                                             1,
                                                                           ),
                                                                    )), 0) - j >
                                                     0,
                                                     j,
                                                   ),
                                         ),
                                 )) - 1

Examples

Basic #

the word $110100111$

live ↗

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

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St001371

FindStat's value for 1

live ↗

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

Known value · FindStat St001371

FindStat's value for 00

live ↗

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

Known value · FindStat St001371

About

Examples	4
Crosswalk	FindStat