WeaklyBelowDyckPathCount

The Dyck paths of the same semilength weakly below this path, itself included.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000418via DyckPath✓↗

WeaklyBelowDyckPathCount(path)The Dyck paths of the same semilength weakly below this path, itself included.

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.
• Weakly below: each entry of the area sequence is at most this path's.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

1 if Length(_x) / 2 < 2 else Apply(
                                    (s: list<integer>) => Max(Sum(Fold(
                                                                        Function(Map(
                                                                                      Function(Sum(Map(
                                                                                                        Function(a[u], u),
                                                                                                        Range(Max([w - 1, 1]), Length(_x) / 2),
                                                                                                      )) if w - 1 <=
                                                                                                           b else 0, w),
                                                                                      Range(1, Length(_x) / 2),
                                                                                    ), a, b),
                                                                        Map(
                                                                             Function(
                                                                                       1 if w == 1 else 0,
                                                                                       w,
                                                                                     ),
                                                                             Range(1, Length(_x) / 2),
                                                                           ),
                                                                        Map(
                                                                             Function(s[i], i),
                                                                             Range(2, Length(_x) / 2),
                                                                           ),
                                                                      )), 0),
                                    Map(
                                         Function(0 if i < 2 else Sum(Map(
                                                                           Function(2 * _x[k] -
                                                                                    1, k),
                                                                           Range(1, i - 1),
                                                                         )), i),
                                         Filter(
                                                 Range(1, Length(_x)),
                                                 Function(_x[i] == 1, i),
                                               ),
                                       ),
                                  )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000418

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

live ↗

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

Known value · FindStat St000418

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

live ↗

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

Known value · FindStat St000418

About

Examples	5
Crosswalk	FindStat