BorderCount

The prefixes shorter than the word that are also suffixes, the empty one included.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001884via BinaryWord✓↗

BorderCount(word)The prefixes shorter than the word that are also suffixes, the empty one included.

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 St001884.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 1 else 1 if Length(_x) < 2 else 1 + Count(Filter(
                                                                    Range(
                                                                           1,
                                                                           Length(_x) -
                                                                           1,
                                                                           1,
                                                                         ),
                                                                    Function(Max(Sum(Map(
                                                                                          Function((_x[1 +
                                                                                                       (m - 1)] +
                                                                                                    1) *
                                                                                                   3 ^
                                                                                                   (l - m), m),
                                                                                          Range(1, l, 1),
                                                                                        )), 0) ==
                                                                             Max(
                                                                                  Sum(Map(Function((_x[Length(_x) - l +
                                                                                                       1 +
                                                                                                       (m - 1)] +
                                                                                                    1) *
                                                                                                   3 ^
                                                                                                   (l - m), m), Range(1, l, 1))),
                                                                                  0,
                                                                                ), l),
                                                                  ))

Examples

Basic #

the word $110100111$

live ↗

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

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St001884

FindStat's value for 1

live ↗

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

Known value · FindStat St001884

FindStat's value for 00

live ↗

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

Known value · FindStat St001884

About

Examples	4
Crosswalk	FindStat