PeakLeftToRightMaxima

Peaks higher than every peak to their left.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001732via DyckPath✓↗

PeakLeftToRightMaxima(path)Peaks higher than every peak to their left.

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

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001732

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

live ↗

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

Known value · FindStat St001732

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

live ↗

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

Known value · FindStat St001732

About

Examples	5
Crosswalk	FindStat