UniqueFactorLength

The least length for which all factors of that length are different.

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UniqueFactorLength(word)The least length for which all factors of that length are different.

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 St000922.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 1 else Min(Filter(
                                     Range(1, Length(_x)),
                                     Function(Length(Unique(Map(
                                                                 Function(Max(Sum(Map(
                                                                                       Function((_x[i +
                                                                                                    (m - 1)] +
                                                                                                 1) *
                                                                                                3 ^
                                                                                                (l - m), m),
                                                                                       Range(1, l, 1),
                                                                                     )), 0), i),
                                                                 Range(
                                                                        1,
                                                                        Length(_x) -
                                                                        l + 1,
                                                                        1,
                                                                      ),
                                                               ))) == Length(_x) -
                                                                     l + 1, l),
                                   ))

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{UniqueFactorLength}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$3$

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St000922

FindStat's value for 1

live ↗

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

Known value · FindStat St000922

FindStat's value for 00

live ↗

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

Known value · FindStat St000922

About

Examples	4
Crosswalk	FindStat