DualRegularModuleFirstExtensionDimension

The dimension of the first extensions of the direct sum of the indecomposable injective modules by the regular module.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000951via DyckPath✓↗

DualRegularModuleFirstExtensionDimension(path)The dimension of the first extensions of the direct sum of the indecomposable injective modules 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 St000951: the dimension of Ext^1(D(A), A). Ext^1 of an interval module is read off the maps from its syzygy and its projective cover.

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(Apply(
                                                                                       (top: integer) => Max(Sum(Map(Function(Max(0, Min(1, w - 1 +
                                                                                                                                            series[w] -
                                                                                                                                            (top +
                                                                                                                                             (v - top)), Min(series[top + 1] -
                                                                                                                                                  (v - top), series[w]) -
                                                                                                                                              (w - 1 +
                                                                                                                                               series[w] -
                                                                                                                                               (top +
                                                                                                                                                (v - top))) +
                                                                                                                                              1)) -
                                                                                                                              Max(0, Min(1, w - 1 +
                                                                                                                                            series[w] -
                                                                                                                                            top, Min(series[top + 1], series[w]) -
                                                                                                                                              (w - 1 +
                                                                                                                                               series[w] -
                                                                                                                                               top) +
                                                                                                                                              1)) +
                                                                                                                              Max(0, Min(1, w - 1 +
                                                                                                                                            series[w] -
                                                                                                                                            top, Min(v - top, series[w]) -
                                                                                                                                              (w - 1 +
                                                                                                                                               series[w] -
                                                                                                                                               top) +
                                                                                                                                              1)), w), Range(1, Length(_x) / 2 +
                                                                                                                                             1))), 0),
                                                                                       Min(Filter(
                                                                                                   Range(0, v - 1, 1),
                                                                                                   Function(x +
                                                                                                            series[x + 1] >
                                                                                                            v - 1, x),
                                                                                                 )),
                                                                                     ), 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$\mathrm{DualRegularModuleFirstExtensionDimension}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$2$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000951

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

live ↗

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

Known value · FindStat St000951

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

live ↗

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

Known value · FindStat St000951

About

Examples	5
Crosswalk	FindStat