InjectivesOfGradeAtLeastOne

The indecomposable injective modules of grade at least one: Hom into the regular module vanishes.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001184via DyckPath✓↗

InjectivesOfGradeAtLeastOne(path)The indecomposable injective modules of grade at least one: Hom into the regular module vanishes.

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 St001184. The grade of a module M is the least i with Ext^i(M, A) nonzero.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (series: list<integer>) => Count(Filter(Map(Function(Apply(
                                                                                            (top: integer) => Fold(
                                                                                                                    Function(Apply((cursor: integer) => Apply((extent: integer) => Apply(
                                                                                                                                                                                          (extension: integer) => Apply((entry: integer) => Apply((found: integer) => [
                                                                                                                                                                                                                                                                        cursor if extent == entry else cursor + extent,
                                                                                                                                                                                                                                                                        extent if extent == entry else entry - extent,
                                                                                                                                                                                                                                                                        a[3] if a[4] == 1 else a[5] + 1 if extension > 0 else a[3],
                                                                                                                                                                                                                                                                        found,
                                                                                                                                                                                                                                                                        a[5] + 1,
                                                                                                                                                                                                                                                                      ], 1 if a[4] == 1 ||
                                                                                                                                                                                                                                                                             extension > 0 else 0), series[cursor + 1]),
                                                                                                                                                                                          Max(Sum(Map(Function(Max(0, Min(1, w - 1 +
                                                                                                                                                                                                                             series[w] -
                                                                                                                                                                                                                             (cursor + extent), Min(series[cursor + 1] -
                                                                                                                                                                                                                                   extent, series[w]) -
                                                                                                                                                                                                                               (w - 1 +
                                                                                                                                                                                                                                series[w] -
                                                                                                                                                                                                                                (cursor + extent)) +
                                                                                                                                                                                                                               1)) -
                                                                                                                                                                                                               Max(0, Min(1, w - 1 +
                                                                                                                                                                                                                             series[w] -
                                                                                                                                                                                                                             cursor, Min(series[cursor + 1], series[w]) -
                                                                                                                                                                                                                               (w - 1 +
                                                                                                                                                                                                                                series[w] -
                                                                                                                                                                                                                                cursor) +
                                                                                                                                                                                                                               1)) +
                                                                                                                                                                                                               Max(0, Min(1, w - 1 +
                                                                                                                                                                                                                             series[w] -
                                                                                                                                                                                                                             cursor, Min(extent, series[w]) -
                                                                                                                                                                                                                               (w - 1 +
                                                                                                                                                                                                                                series[w] -
                                                                                                                                                                                                                                cursor) +
                                                                                                                                                                                                                               1)), w), Range(1, Length(_x) / 2 +
                                                                                                                                                                                                                              1))), 0),
                                                                                                                                                                                        ), a[2]), a[1]), a, b),
                                                                                                                    [
                                                                                                                      top,
                                                                                                                      v - top,
                                                                                                                      0,
                                                                                                                      1 if Max(Sum(Map(
                                                                                                                                        Function(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) >
                                                                                                                           0 else 0,
                                                                                                                      0,
                                                                                                                    ],
                                                                                                                    Range(1, Length(_x) / 2 +
                                                                                                                             1),
                                                                                                                  )[3],
                                                                                            Min(Filter(
                                                                                                        Range(0, v - 1, 1),
                                                                                                        Function(x +
                                                                                                                 series[x + 1] >
                                                                                                                 v - 1, x),
                                                                                                      )),
                                                                                          ), v), Range(1, Length(_x) / 2 +
                                                                                                         1)), Function(d >= 1, d))),
                                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$\mathrm{InjectivesOfGradeAtLeastOne}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001184

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

live ↗

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

Known value · FindStat St001184

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

live ↗

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

Known value · FindStat St001184

About

Examples	5
Crosswalk	FindStat