CriticalStepCount

The critical steps l + m in the Catalan factorization (D 0)^l D (1 D)^m.

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CriticalStepCount(word)The critical steps l + m in the Catalan factorization (D 0)^l D (1 D)^m.

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 St000885. The word ends at height m - l, and l is the least height reached below 0.

Definition

Epsil, as the implementation runs it.

on BinaryWord, over _x

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

Examples

Basic #

the word $110100111$

live ↗

In$\mathrm{CriticalStepCount}({\operatorname{BinaryWord}}([1, 1, 0, 1, 0, 0, 1, 1, 1]))$Out$3$

Properties #

FindStat's value for 0

live ↗

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

Known value · FindStat St000885

FindStat's value for 1

live ↗

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

Known value · FindStat St000885

FindStat's value for 00

live ↗

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

Known value · FindStat St000885

About

Examples	4
Crosswalk	FindStat