Balance

The least q such that any two factors of one length differ by at most q ones.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000628via BinaryWord✓↗

Balance(word)The least q such that any two factors of one length differ by at most q ones.

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 St000628.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 1 else Max(Map(Function(Apply(
                                                 (c: list<integer>) => Max(c) -
                                                                       Min(c),
                                                 Map(
                                                      Function(Max(Sum(Map(
                                                                            Function(_x[j], j),
                                                                            Range(
                                                                                   i,
                                                                                   i + l -
                                                                                   1,
                                                                                   1,
                                                                                 ),
                                                                          )), 0), i),
                                                      Range(1, Length(_x) - l +
                                                               1, 1),
                                                    ),
                                               ), l), Range(1, Length(_x), 1)))

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{Balance}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$2$

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St000628

FindStat's value for 1

live ↗

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

Known value · FindStat St000628

FindStat's value for 00

live ↗

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

Known value · FindStat St000628

About

Examples	4
Crosswalk	FindStat