SocleToRadicalExtensionDimension

The dimension of the first extensions of the socle of the regular module by the Jacobson radical.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001227via DyckPath✓↗

SocleToRadicalExtensionDimension(path)The dimension of the first extensions of the socle of the regular module by the Jacobson radical.

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 St001227.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (series: list<integer>) => Apply(
                                                                  (projectiveSocleTops: list<integer>) => Apply(
                                                                                                                 (projectiveSocleLengths: list<integer>) => Apply(
                                                                                                                                                                   (radicalTops: list<integer>) => Apply(
                                                                                                                                                                                                          (radicalLengths: list<integer>) => Max(Sum(Map(Function(Max(Sum(Map(Function(Max(0, Min(1, radicalTops[j] + radicalLengths[j] -
                                                                                                                                                                                                                                                                                                     (projectiveSocleTops[i] + projectiveSocleLengths[i]), Min(series[projectiveSocleTops[i] + 1] -
                                                                                                                                                                                                                                                                                                           projectiveSocleLengths[i], radicalLengths[j]) -
                                                                                                                                                                                                                                                                                                       (radicalTops[j] + radicalLengths[j] -
                                                                                                                                                                                                                                                                                                        (projectiveSocleTops[i] + projectiveSocleLengths[i])) +
                                                                                                                                                                                                                                                                                                       1)) -
                                                                                                                                                                                                                                                                                       Max(0, Min(1, radicalTops[j] + radicalLengths[j] -
                                                                                                                                                                                                                                                                                                     projectiveSocleTops[i], Min(series[projectiveSocleTops[i] + 1], radicalLengths[j]) -
                                                                                                                                                                                                                                                                                                       (radicalTops[j] + radicalLengths[j] -
                                                                                                                                                                                                                                                                                                        projectiveSocleTops[i]) +
                                                                                                                                                                                                                                                                                                       1)) +
                                                                                                                                                                                                                                                                                       Max(0, Min(1, radicalTops[j] + radicalLengths[j] -
                                                                                                                                                                                                                                                                                                     projectiveSocleTops[i], Min(projectiveSocleLengths[i], radicalLengths[j]) -
                                                                                                                                                                                                                                                                                                       (radicalTops[j] + radicalLengths[j] -
                                                                                                                                                                                                                                                                                                        projectiveSocleTops[i]) +
                                                                                                                                                                                                                                                                                                       1)), j), Range(1, Length(radicalTops)))), 0), i), Range(1, Length(projectiveSocleTops)))), 0),
                                                                                                                                                                                                          Map(Function(series[v] - 1, v), Range(1, Length(_x) / 2)),
                                                                                                                                                                                                        ),
                                                                                                                                                                   Map(Function(v, v), Range(1, Length(_x) / 2)),
                                                                                                                                                                 ),
                                                                                                                 Map(
                                                                                                                      Function(1, w),
                                                                                                                      Range(1, Length(_x) / 2 +
                                                                                                                               1),
                                                                                                                    ),
                                                                                                               ),
                                                                  Map(
                                                                       Function(w + series[w] -
                                                                                2, w),
                                                                       Range(
                                                                              1,
                                                                              Length(_x) /
                                                                              2 +
                                                                              1,
                                                                            ),
                                                                     ),
                                                                ),
                                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$\mathrm{SocleToRadicalExtensionDimension}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001227

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

live ↗

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

Known value · FindStat St001227

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

live ↗

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

Known value · FindStat St001227

About

Examples	5
Crosswalk	FindStat