CenteredMultitunnels

Factorizations of the path into three parts, the middle one a Dyck path and the outer two of equal length.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000675via DyckPath✓↗

CenteredMultitunnels(path)Factorizations of the path into three parts, the middle one a Dyck path and the outer two of equal length.

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 Apply(
                                (v: list<integer>) => Count(Filter(
                                                                    Range(
                                                                           1,
                                                                           Length(_x) /
                                                                           2,
                                                                         ),
                                                                    Function(v[Length(_x) /
                                                                               2 +
                                                                               c +
                                                                               1] ==
                                                                             v[Length(_x) /
                                                                               2 -
                                                                               c +
                                                                               1] &&
                                                                             Min(Map(
                                                                                      Function(v[j], j),
                                                                                      Range(Length(_x) / 2 -
                                                                                            c +
                                                                                            1, Length(_x) / 2 +
                                                                                              c + 1),
                                                                                    )) >=
                                                                             v[Length(_x) /
                                                                               2 -
                                                                               c +
                                                                               1], c),
                                                                  )),
                                Join(
                                      [0],
                                      Map(
                                           Function(
                                                     Sum(Map(
                                                              Function(
                                                                        2 *
                                                                        _x[k] - 1,
                                                                        k,
                                                                      ),
                                                              Range(1, i),
                                                            )),
                                                     i,
                                                   ),
                                           Range(1, Length(_x)),
                                         ),
                                    ),
                              )

Examples

Basic #

the step word $UUDDUD$

live ↗

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

the step word $UDUUDD$

live ↗

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

Properties #

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

live ↗

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

Known value · FindStat St000675

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

live ↗

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

Known value · FindStat St000675

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

live ↗

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

Known value · FindStat St000675

About

Examples	5
Crosswalk	FindStat