Hills

Peaks at height 1 — an up step from the axis immediately followed by a down step.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000674via DyckPath✓↗

Hills(path)Peaks at height 1 — an up step from the axis immediately followed by a down step.

Typed signatures

Type	Package	Applies
(dyck_path | schroeder_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.
• Hills of size one: a hill of size k is k up steps then k down steps from the axis (HillsOfSizeTwo, HillsOfSizeThree).

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 2 else Count(Filter(
                                       Range(1, Length(_x) - 1),
                                       Function(
                                                 _x[i] == 1 && _x[i + 1] == 0 &&
                                                 Sum(Map(
                                                          Function(
                                                                    2 * _x[k] -
                                                                    1,
                                                                    k,
                                                                  ),
                                                          Range(1, i),
                                                        )) == 1,
                                                 i,
                                               ),
                                     ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

the Schroeder path $UUDDUDL^2$: only $UD$ from the axis is a hill

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SchroederPath}}([1, 1, -1, -1, 1, -1, 2]), \text{Hills})$Out$1$

Properties #

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

live ↗

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

Known value · FindStat St000674

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

live ↗

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

Known value · FindStat St000674

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 0, 1, 0, 1, 0]), \text{Hills})$Out$3$

Known value · FindStat St000674

About

Examples	6
Crosswalk	FindStat