RepetitionExponent

The largest e such that the word is e copies of one word.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000627via BinaryWord✓↗

RepetitionExponent(word)The largest e such that the word is e copies of one word.

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 St000627, also called the frequency. The word is copies of a word of length q when q divides n and the first n - q letters are the last n - q.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 1 else 1 if Length(_x) < 2 else Length(_x) / Min(Join(
                                                                         [Length(_x)],
                                                                         Filter(
                                                                                 Range(1, Length(_x) - 1, 1),
                                                                                 Function(Max(Sum(Map(
                                                                                                       Function((_x[1 +
                                                                                                                    (m - 1)] +
                                                                                                                 1) *
                                                                                                                3 ^
                                                                                                                (Length(_x) - q -
                                                                                                                 m), m),
                                                                                                       Range(1, Length(_x) - q, 1),
                                                                                                     )), 0) ==
                                                                                          Max(Sum(Map(
                                                                                                       Function((_x[Length(_x) -
                                                                                                                    (Length(_x) - q) +
                                                                                                                    1 +
                                                                                                                    (m - 1)] +
                                                                                                                 1) *
                                                                                                                3 ^
                                                                                                                (Length(_x) - q -
                                                                                                                 m), m),
                                                                                                       Range(1, Length(_x) - q, 1),
                                                                                                     )), 0) if Length(_x) % q ==
                                                                                              0 else False, q),
                                                                               ),
                                                                       ))

Examples

Basic #

the word $110100111$

live ↗

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

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St000627

FindStat's value for 1

live ↗

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

Known value · FindStat St000627

FindStat's value for 00

live ↗

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

Known value · FindStat St000627

About

Examples	4
Crosswalk	FindStat