InteractionDifference

The pyramid weight less the upper interactions: lower interactions less upper interactions.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000335via DyckPath✓↗

InteractionDifference(path)The pyramid weight less the upper interactions: lower interactions less upper interactions.

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

Fold(Function(
               [a[1], 1 if a[3] == 0 else a[2] + 1, 1] if b == 1 else [
                                                                        a[1] + 1,
                                                                        a[2] - 1,
                                                                        0,
                                                                      ] if a[2] >
                                                                          0 else [a[1], a[2], 0],
               a,
               b,
             ), [0, 0, 0], _x)[1] - 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$\mathrm{InteractionDifference}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$2$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000335

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

live ↗

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

Known value · FindStat St000335

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

live ↗

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

Known value · FindStat St000335

About

Examples	5
Crosswalk	FindStat