DistinctCubeCount

The distinct nonempty words u with uuu a factor.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001423via BinaryWord✓↗

DistinctCubeCount(word)The distinct nonempty words u with uuu a factor.

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 St001423.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 3 else Max(Sum(Map(Function(Max(Sum(Map(
                                                           Function(
                                                                     1 if n > 0 else 0,
                                                                     n,
                                                                   ),
                                                           Unique(Map(Function(Max(
                                                                                    Sum(Map(Function((_x[i +
                                                                                                         (m - 1)] +
                                                                                                      1) *
                                                                                                     3 ^
                                                                                                     (l - m), m), Range(1, l, 1))),
                                                                                    0,
                                                                                  ) if Max(Sum(Map(Function((_x[i +
                                                                                                               (m - 1)] +
                                                                                                            1) *
                                                                                                           3 ^
                                                                                                           (l - m), m), Range(1, l, 1))), 0) ==
                                                                                      Max(Sum(Map(Function((_x[i + 1l +
                                                                                                               (m - 1)] +
                                                                                                            1) *
                                                                                                           3 ^
                                                                                                           (l - m), m), Range(1, l, 1))), 0) &&
                                                                                      Max(Sum(Map(Function((_x[i +
                                                                                                               (m - 1)] +
                                                                                                            1) *
                                                                                                           3 ^
                                                                                                           (l - m), m), Range(1, l, 1))), 0) ==
                                                                                      Max(Sum(Map(Function((_x[i + 2l +
                                                                                                               (m - 1)] +
                                                                                                            1) *
                                                                                                           3 ^
                                                                                                           (l - m), m), Range(1, l, 1))), 0) else 0, i), Range(1, Length(_x) - 3l +
                                                                                                            1, 1))),
                                                         )), 0), l), Range(
                                                                           1,
                                                                           (Length(_x) -
                                                                            Length(_x) %
                                                                            3) /
                                                                           3,
                                                                           1,
                                                                         ))), 0)

Examples

Basic #

the word $110100111$

live ↗

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

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St001423

FindStat's value for 1

live ↗

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

Known value · FindStat St001423

FindStat's value for 00

live ↗

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

Known value · FindStat St001423

About

Examples	4
Crosswalk	FindStat