MinimalValleyHeight

The lowest height of a valley, or the semilength if the path has none.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001107via DyckPath✓↗

MinimalValleyHeight(path)The lowest height of a valley, or the semilength if the path has none.

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) < 2 else Apply(
                                (valleyHeights: list<integer>) => Length(_x) / 2 if Length(valleyHeights) == 0 else Min(valleyHeights),
                                Map(
                                     Function(Sum(Map(
                                                       Function(
                                                                 2 * _x[k] - 1,
                                                                 k,
                                                               ),
                                                       Range(1, i),
                                                     )), i),
                                     Filter(
                                             Range(1, Length(_x) - 1),
                                             Function(_x[i] == 0 && _x[i + 1] ==
                                                                    1, i),
                                           ),
                                   ),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St001107

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 1, 0, 0]), \text{MinimalValleyHeight})$Out$2$

Known value · FindStat St001107

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

live ↗

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

Known value · FindStat St001107

About

Examples	5
Crosswalk	FindStat