ShortestPalindromicDecompositionLength

The fewest palindromic factors the word is a concatenation of.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000630via BinaryWord✓↗

ShortestPalindromicDecompositionLength(word)The fewest palindromic factors the word is a concatenation of.

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 St000630. The table holds the fewest pieces for each prefix.

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(
                                                                                a,
                                                                                [Min(Map(
                                                                                          Function(a[i + 1] +
                                                                                                   1 if Max(Sum(Map(Function((_x[i + 1 +
                                                                                                                                (m - 1)] +
                                                                                                                             1) *
                                                                                                                            3 ^
                                                                                                                            (b - i -
                                                                                                                             m), m), Range(1, b - i, 1))), 0) ==
                                                                                                       Max(Sum(Map(Function((_x[i + 1 +
                                                                                                                                (m - 1)] +
                                                                                                                             1) *
                                                                                                                            3 ^
                                                                                                                            (m - 1), m), Range(1, b - i, 1))), 0) else Length(_x) + 1, i),
                                                                                          Range(0, b - 1, 1),
                                                                                        ))],
                                                                              ),
                                                                        ), a_init, Range(1, Length(_x))), [0])[Length(_x) + 1]

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{ShortestPalindromicDecompositionLength}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$4$

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St000630

FindStat's value for 1

live ↗

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

Known value · FindStat St000630

FindStat's value for 00

live ↗

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

Known value · FindStat St000630

About

Examples	4
Crosswalk	FindStat