Dinv

The dinv statistic, read from the area sequence.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000006via DyckPath✓↗

Dinv(path)The dinv statistic, read from the area sequence.

SageMathenumeratio_stat("DyckPath", path, "Dinv")

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.
• dinv = #{i<j : a_i=a_j} + #{i<j : a_i=a_j+1}; the two counts never overlap (a_i=a_j and a_i=a_j+1 can't both hold), so this is one pass over the pairs.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000006

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

live ↗

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

Known value · FindStat St000006

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

live ↗

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

Known value · FindStat St000006

About

Examples	5
Checked against	SageMath 5 agree
Crosswalk	FindStatSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts