DinvDeficit

The binomial(n, 2) less the area and the dinv: the boxes of the partition traced out by the path where leg minus arm is not 0 or 1.

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DinvDeficit(path)The binomial(n, 2) less the area and the dinv: the boxes of the partition traced out by the path where leg minus arm is not 0 or 1.

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) < 4 else Sum(Map(Function(Count(Filter(
                                                                                  Range(i + 1, Length(_x) / 2),
                                                                                  Function(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),
                                                                                                      ),
                                                                                              )[i] ==
                                                                                           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),
                                                                                                      ),
                                                                                              )[j] ||
                                                                                           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),
                                                                                                      ),
                                                                                              )[i] ==
                                                                                           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),
                                                                                                      ),
                                                                                              )[j] +
                                                                                           1, j),
                                                                                )) if i <
                                                                                     Length(_x) / 2 else 0, i), Range(1, Length(_x) / 2))))

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000369

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

live ↗

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

Known value · FindStat St000369

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

live ↗

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

Known value · FindStat St000369

About

Examples	5
Crosswalk	FindStat