SocleSimplesOfVanishingFirstExtensionWithRegularModule

The simple modules in the socle of the regular module that have no first extension by the regular module.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001483via DyckPath✓↗

SocleSimplesOfVanishingFirstExtensionWithRegularModule(path)The simple modules in the socle of the regular module that have no first extension by the regular module.

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.
• FindStat's St001483. A simple module is in the socle when some indecomposable projective ends at it.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (series: list<integer>) => Max(Sum(Map(Function(Min(
                                                                                     1,
                                                                                     Max(Sum(Map(
                                                                                                  Function(1 if w + series[w] ==
                                                                                                                v + 1 else 0, w),
                                                                                                  Range(1, Length(_x) / 2 +
                                                                                                           1),
                                                                                                )), 0),
                                                                                   ) *
                                                                                (1 if Max(Sum(Map(Function(Max(0, Min(
                                                                                                                       1,
                                                                                                                       w - 1 +
                                                                                                                       series[w] -
                                                                                                                       (v - 1 +
                                                                                                                        1),
                                                                                                                       Min(series[v - 1 +
                                                                                                                                  1] -
                                                                                                                           1, series[w]) -
                                                                                                                       (w - 1 +
                                                                                                                        series[w] -
                                                                                                                        (v - 1 +
                                                                                                                         1)) +
                                                                                                                       1,
                                                                                                                     )) -
                                                                                                           Max(0, Min(1, w - 1 +
                                                                                                                         series[w] -
                                                                                                                         (v - 1), Min(series[v - 1 +
                                                                                                                                      1], series[w]) -
                                                                                                                           (w - 1 +
                                                                                                                            series[w] -
                                                                                                                            (v - 1)) +
                                                                                                                           1)) +
                                                                                                           Max(0, Min(1, w - 1 +
                                                                                                                         series[w] -
                                                                                                                         (v - 1), Min(1, series[w]) -
                                                                                                                           (w - 1 +
                                                                                                                            series[w] -
                                                                                                                            (v - 1)) +
                                                                                                                           1)), w), Range(1, Length(_x) / 2 +
                                                                                                                          1))), 0) ==
                                                                                      0 else 0), v), Range(1, Length(_x) / 2 +
                                                                                               1))), 0),
                                Apply(
                                       (h: list<integer>) => Apply(
                                                                    (ups: list<integer>) => Join(Map(
                                                                                                      Function(h[ups[Length(_x) / 2 +
                                                                                                                     1 -
                                                                                                                     t]] +
                                                                                                               1, t),
                                                                                                      Range(1, Length(_x) / 2),
                                                                                                    ), [1]),
                                                                    Filter(
                                                                            Range(1, Length(_x)),
                                                                            Function(_x[i] == 1, i),
                                                                          ),
                                                                  ),
                                       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$\mathrm{SocleSimplesOfVanishingFirstExtensionWithRegularModule}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001483

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

live ↗

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

Known value · FindStat St001483

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

live ↗

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

Known value · FindStat St001483

About

Examples	5
Crosswalk	FindStat