LogarithmicHeight

The floor of the binary logarithm of the height plus one.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000920via DyckPath✓↗

LogarithmicHeight(path)The floor of the binary logarithm of the height plus one.

Typed signatures

Type	Package	Applies
(dyck_path) -> number	combinatorics	always

Domain: Dyck path statistics

Details

• Defined over DyckPath as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a DyckPath 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.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (h: number) => Count(Filter(
                                                             Range(1, h + 1),
                                                             Function(
                                                                       2 ^ k <=
                                                                       h + 1,
                                                                       k,
                                                                     ),
                                                           )),
                                Max(Map(Function(
                                                  Sum(Map(
                                                           Function(2 * _x[k] -
                                                                    1, k),
                                                           Range(1, i),
                                                         )),
                                                  i,
                                                ), Range(1, Length(_x)))),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{LogarithmicHeight}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

In$\mathrm{LogarithmicHeight}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$1$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000920

FindStat's value for [1,0,1,0]

live ↗

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

Known value · FindStat St000920

FindStat's value for [1,1,0,0]

live ↗

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

Known value · FindStat St000920

About

Examples	5
Crosswalk	FindStat