BoxWeight

The product, over the up steps, of one more than the number of down steps before it.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001808via DyckPath✓↗

BoxWeight(path)The product, over the up steps, of one more than the number of down steps before it.

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.
• Follows FindStat's Sage code and its data, not its description: the description gives 4 for the path UUDD, the code and every stored value give 1.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

1 if Length(_x) < 1 else Product(Map(
                                      Function(1 + Count(Filter(
                                                                 Range(1, i),
                                                                 Function(_x[d] ==
                                                                          0, d),
                                                               )), i),
                                      Filter(
                                              Range(1, Length(_x)),
                                              Function(_x[i] == 1, i),
                                            ),
                                    ))

Examples

Basic #

the step word $UUDDUD$

live ↗

In$\mathrm{BoxWeight}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$3$

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St001808

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

live ↗

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

Known value · FindStat St001808

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

live ↗

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

Known value · FindStat St001808

About

Examples	5
Crosswalk	FindStat