SameEndpointPathsAboveWordCount

The paths with the word's number of ones and zeros that stay weakly above the path the word gives.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001313via BinaryWord✓↗

SameEndpointPathsAboveWordCount(word)The paths with the word's number of ones and zeros that stay weakly above the path the word gives.

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 St001313, the count of St001365 that end where the word does: ahead by 0 steps up.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 1 else Apply(
                                (a_init: list<integer>) => Fold((
                                                                  a: list<integer>,
                                                                  b,
                                                                ) => Apply(
                                                                           (a_next: list<integer>) => a_next,
                                                                           Join(
                                                                                 [0],
                                                                                 Map(
                                                                                      Function(a[j] +
                                                                                               a[j + 1] if b == 1 else a[j - 1] +
                                                                                                               a[j], j),
                                                                                      Range(2, Length(_x) + 2, 1),
                                                                                    ),
                                                                                 [0],
                                                                               ),
                                                                         ), a_init, _x),
                                Join([0, 1], Map(
                                                  Function(0, z),
                                                  Range(1, Length(_x) + 1, 1),
                                                )),
                              )[2]

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{SameEndpointPathsAboveWordCount}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$30$

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St001313

FindStat's value for 1

live ↗

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

Known value · FindStat St001313

FindStat's value for 00

live ↗

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

Known value · FindStat St001313

About

Examples	4
Crosswalk	FindStat