BounceDeficit

The binomial(n, 2) less the area and the bounce.

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BounceDeficit(path)The binomial(n, 2) less the area and the bounce.

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

0 if Length(_x) < 1 else Length(_x) / 2 * (Length(_x) / 2 - 1) / 2 -
                         (0 if Length(_x) < 1 else (Sum(Map(
                                                             Function(Sum(Map(
                                                                               Function(2 * _x[k] -
                                                                                        1, k),
                                                                               Range(1, i),
                                                                             )), i),
                                                             Range(
                                                                    1,
                                                                    Length(_x),
                                                                  ),
                                                           )) - Length(_x) / 2) /
                                                   2) - (0 if Length(_x) < 2 else Apply(
                                                             (ceilings: list<integer>) => Fold(
                                                                                                Function([a[1], a[2], a[3]] if a[1] ==
                                                                                                                               Length(_x) / 2 else [
                                                                                                                                       ceilings[a[1] + 1],
                                                                                                                                       a[2] + 1,
                                                                                                                                       a[3] +
                                                                                                                                       a[2] *
                                                                                                                                       (ceilings[a[1] + 1] -
                                                                                                                                        a[1]),
                                                                                                                                     ], a, b),
                                                                                                [0, 0, 0],
                                                                                                Range(1, Length(_x) / 2),
                                                                                              )[3],
                                                             Apply(
                                                                    (ups: list<integer>) => Map(Function(Sum(Map(
                                                                                                                  Function(2 * _x[k] -
                                                                                                                           1, k),
                                                                                                                  Range(1, ups[Length(_x) / 2 +
                                                                                                                               1 -
                                                                                                                               j]),
                                                                                                                )) +
                                                                                                         (j - 1), j), Range(1, Length(_x) / 2)),
                                                                    Filter(
                                                                            Range(1, Length(_x)),
                                                                            Function(_x[i] == 1, i),
                                                                          ),
                                                                  ),
                                                           ))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000376

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

live ↗

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

Known value · FindStat St000376

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

live ↗

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

Known value · FindStat St000376

About

Examples	5
Crosswalk	FindStat