WeaklyAboveLatticePathCount

The up-and-down paths of the same length, ending anywhere, that stay weakly above this path.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001361via DyckPath✓↗

WeaklyAboveLatticePathCount(path)The up-and-down paths of the same length, ending anywhere, that stay weakly above this path.

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.
• The paths need not return to the axis: 2^n for the highest path, the central binomial coefficient for the zigzag.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

1 if Length(_x) < 1 else Apply((m: integer) => Apply(
                                                      (start: list<integer>) => Max(Sum(Fold((a: list<integer>, b) => Apply(
                                                                                                                             (next: list<integer>) => next,
                                                                                                                             Map(Function(a[w] +
                                                                                                                                          (a[w + 1] if w < m else 0) if b == 1 else a[w] +
                                                                                                                                                          (a[w - 1] if w > 1 else 0), w), Range(1, m)),
                                                                                                                           ), start, _x)), 0),
                                                      Map(Function(
                                                                    1 if w == 1 else 0,
                                                                    w,
                                                                  ), Range(1, m)),
                                                    ), Length(_x) + 1)

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{WeaklyAboveLatticePathCount}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$14$

the step word $UDUUDD$

live ↗

In$\mathrm{WeaklyAboveLatticePathCount}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$16$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001361

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 0, 1, 0]), \text{WeaklyAboveLatticePathCount})$Out$6$

Known value · FindStat St001361

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 1, 0, 0]), \text{WeaklyAboveLatticePathCount})$Out$4$

Known value · FindStat St001361

About

Examples	5
Crosswalk	FindStat