OddPalindromicPrefixHalfLength

The largest j such that the prefix of length 2 j + 1 is a palindrome.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001414via BinaryWord✓↗

OddPalindromicPrefixHalfLength(word)The largest j such that the prefix of length 2 j + 1 is a palindrome.

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 St001414.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

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

Examples

Basic #

the word $110100111$

live ↗

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

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St001414

FindStat's value for 1

live ↗

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

Known value · FindStat St001414

FindStat's value for 00

live ↗

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

Known value · FindStat St001414

About

Examples	4
Crosswalk	FindStat