UpperInteractions

Over each valley, the shorter of the down run before it and the up run after it, summed: the factors of k down steps then k up steps.

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UpperInteractions(path)Over each valley, the shorter of the down run before it and the up run after it, summed: the factors of k down steps then k up steps.

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.
• Over a valley, the shorter of the down run before it and the up run after it.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

Fold(Function(
               [a[1], 1 if a[3] == 1 else a[2] + 1, 0] if b == 0 else [
                                                                        a[1] + 1,
                                                                        a[2] - 1,
                                                                        1,
                                                                      ] if a[2] >
                                                                          0 else [a[1], a[2], 1],
               a,
               b,
             ), [0, 0, 0], _x)[1]

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000331

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

live ↗

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

Known value · FindStat St000331

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

live ↗

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

Known value · FindStat St000331

About

Examples	5
Crosswalk	FindStat