ValleyFillingDegree

The number of times every valley 01 must be swapped for a peak 10, and a trailing 0 made a 1, to leave no zeros.

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ValleyFillingDegree(word)The number of times every valley 01 must be swapped for a peak 10, and a trailing 0 made a 1, to leave no zeros.

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 St001721.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

0 if Length(_x) < 1 else Max(
                              0,
                              Fold(Function(
                                             Max(a + 2, b) if _x[Length(_x) + 1 -
                                                                 b] == 0 else a,
                                             a,
                                             b,
                                           ), -1, Range(1, Length(_x))),
                            )

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{ValleyFillingDegree}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$8$

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St001721

FindStat's value for 1

live ↗

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

Known value · FindStat St001721

FindStat's value for 00

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([0, 0]), \text{ValleyFillingDegree})$Out$3$

Known value · FindStat St001721

About

Examples	4
Crosswalk	FindStat