DecimalRepresentationWithLeadingOne

The word with a 1 put in front, read as a binary numeral.

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DecimalRepresentationWithLeadingOne(word)The word with a 1 put in front, read as a binary numeral.

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 St000827.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

1 if Length(_x) < 1 else 2 ^ Length(_x) + Max(
                                               Sum(Map(
                                                        Function(
                                                                  _x[i] * 2 ^ (Length(_x) -
                                                                               i),
                                                                  i,
                                                                ),
                                                        Range(1, Length(_x)),
                                                      )),
                                               0,
                                             )

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{DecimalRepresentationWithLeadingOne}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$935$

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St000827

FindStat's value for 1

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([1]), \text{DecimalRepresentationWithLeadingOne})$Out$3$

Known value · FindStat St000827

FindStat's value for 00

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([0, 0]), \text{DecimalRepresentationWithLeadingOne})$Out$4$

Known value · FindStat St000827

About

Examples	4
Crosswalk	FindStat