SimplesNotInSocleOrOfVanishingFirstExtensionWithRegularModule

The simple modules that are not in the socle of the regular module or have no first extension by the regular module.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001492via DyckPath✓↗

SimplesNotInSocleOrOfVanishingFirstExtensionWithRegularModule(path)The simple modules that are not in the socle of the regular module or 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 St001492. 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(1 if Max(Sum(Map(
                                                                                                   Function(1 if w + series[w] ==
                                                                                                                 v + 1 else 0, w),
                                                                                                   Range(1, Length(_x) / 2 +
                                                                                                            1),
                                                                                                 )), 0) ==
                                                                                      0 else 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$\mathrm{SimplesNotInSocleOrOfVanishingFirstExtensionWithRegularModule}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$3$

the step word $UDUUDD$

live ↗

In$\mathrm{SimplesNotInSocleOrOfVanishingFirstExtensionWithRegularModule}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$3$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001492

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

live ↗

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

Known value · FindStat St001492

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 1, 0, 0]), \text{SimplesNotInSocleOrOfVanishingFirstExtensionWithRegularModule})$Out$3$

Known value · FindStat St001492

About

Examples	5
Crosswalk	FindStat