SphericalSimpleModules

The simple modules with Ext^i(S, A) zero for 0 < i < the projective dimension of S; those of projective dimension at most one count.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001994via DyckPath✓↗

SphericalSimpleModules(path)The simple modules with Ext^i(S, A) zero for 0 < i < the projective dimension of S; those of projective dimension at most one count.

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 St001994.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (series: list<integer>) => Apply(
                                                                  (dims: list<integer>) => Max(Sum(Map(
                                                                                                        Function(1 if dims[v] <= 1 else 1 if Max(Sum(Map(Function(Apply(
                                                                                                                                                                         (syzygy: list<integer>) => Max(Sum(Map(Function(Max(0, Min(1, w - 1 +
                                                                                                                                                                                                                                       series[w] -
                                                                                                                                                                                                                                       (syzygy[1] + syzygy[2]), Min(series[syzygy[1] + 1] -
                                                                                                                                                                                                                                             syzygy[2], series[w]) -
                                                                                                                                                                                                                                         (w - 1 +
                                                                                                                                                                                                                                          series[w] -
                                                                                                                                                                                                                                          (syzygy[1] + syzygy[2])) +
                                                                                                                                                                                                                                         1)) -
                                                                                                                                                                                                                         Max(0, Min(1, w - 1 +
                                                                                                                                                                                                                                       series[w] -
                                                                                                                                                                                                                                       syzygy[1], Min(series[syzygy[1] + 1], series[w]) -
                                                                                                                                                                                                                                         (w - 1 +
                                                                                                                                                                                                                                          series[w] -
                                                                                                                                                                                                                                          syzygy[1]) +
                                                                                                                                                                                                                                         1)) +
                                                                                                                                                                                                                         Max(0, Min(1, w - 1 +
                                                                                                                                                                                                                                       series[w] -
                                                                                                                                                                                                                                       syzygy[1], Min(syzygy[2], series[w]) -
                                                                                                                                                                                                                                         (w - 1 +
                                                                                                                                                                                                                                          series[w] -
                                                                                                                                                                                                                                          syzygy[1]) +
                                                                                                                                                                                                                                         1)), w), Range(1, Length(_x) / 2 +
                                                                                                                                                                                                                                        1))), 0),
                                                                                                                                                                         Apply((target: integer) => Fold(
                                                                                                                                                                                                          Function(Apply(
                                                                                                                                                                                                                          (entry: integer) => [a[1], a[2], a[3]] if a[2] == entry ||
                                                                                                                                                                                                                                                                    a[3] == target else [a[1] + a[2], entry - a[2], a[3] + 1],
                                                                                                                                                                                                                          series[a[1] + 1],
                                                                                                                                                                                                                        ), a, b),
                                                                                                                                                                                                          [v - 1, 1, 0],
                                                                                                                                                                                                          Range(1, Length(_x) / 2 +
                                                                                                                                                                                                                   1),
                                                                                                                                                                                                        ), i - 1),
                                                                                                                                                                       ) if i < dims[v] else 0, i), Range(1, Length(_x) / 2))), 0) ==
                                                                                                                                             0 else 0, v),
                                                                                                        Range(1, Length(_x) / 2 +
                                                                                                                 1),
                                                                                                      )), 0),
                                                                  Map(
                                                                       Function(Fold(
                                                                                      Function(Apply(
                                                                                                      (entry: integer) => [a[1], a[2], a[3]] if a[2] == entry else [a[1] + a[2], entry - a[2], a[3] + 1],
                                                                                                      series[a[1] + 1],
                                                                                                    ), a, b),
                                                                                      [v - 1, 1, 0],
                                                                                      Range(1, Length(_x) / 2 +
                                                                                               1),
                                                                                    )[3], v),
                                                                       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$\mathrm{SphericalSimpleModules}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$4$

the step word $UDUUDD$

live ↗

In$\mathrm{SphericalSimpleModules}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$4$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001994

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

live ↗

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

Known value · FindStat St001994

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

live ↗

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

Known value · FindStat St001994

About

Examples	5
Crosswalk	FindStat