ModulesOfProjectiveOrInjectiveDimensionAtMostOne

The indecomposable modules whose projective dimension or injective dimension is at most one.

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ModulesOfProjectiveOrInjectiveDimensionAtMostOne(path)The indecomposable modules whose projective dimension or injective dimension is at most one.

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.

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(Sum(Map(Function(Apply(
                                                                                                                                         (projective: integer) => Apply(
                                                                                                                                                                         (injective: integer) => 1 if projective <= 1 ||
                                                                                                                                                                                                      injective <= 1 else 0,
                                                                                                                                                                         Fold(
                                                                                                                                                                               Function(Apply((socle: integer) => Apply(
                                                                                                                                                                                                                         (hull: integer) => [a[1], a[2], a[3]] if a[1] == hull else [hull, a[1] - hull, a[3] + 1],
                                                                                                                                                                                                                         Min(Filter(Range(0, socle, 1), Function(x +
                                                                                                                                                                                                                                                                 series[x + 1] >
                                                                                                                                                                                                                                                                 socle, x))),
                                                                                                                                                                                                                       ), a[1] + a[2] -
                                                                                                                                                                                                                         1), a, b),
                                                                                                                                                                               [v - 1, y, 0],
                                                                                                                                                                               Range(1, Length(_x) / 2 +
                                                                                                                                                                                        1),
                                                                                                                                                                             )[3],
                                                                                                                                                                       ),
                                                                                                                                         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, y, 0],
                                                                                                                                               Range(1, Length(_x) / 2 +
                                                                                                                                                        1),
                                                                                                                                             )[3],
                                                                                                                                       ), y), Range(1, Max(series[v], 1)))), 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$\mathrm{ModulesOfProjectiveOrInjectiveDimensionAtMostOne}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$8$

the step word $UDUUDD$

live ↗

In$\mathrm{ModulesOfProjectiveOrInjectiveDimensionAtMostOne}({\operatorname{DyckPath}}([1, 0, 1, 1, 0, 0]))$Out$8$

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001138

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 0, 1, 0]), \text{ModulesOfProjectiveOrInjectiveDimensionAtMostOne})$Out$5$

Known value · FindStat St001138

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

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{DyckPath}}([1, 1, 0, 0]), \text{ModulesOfProjectiveOrInjectiveDimensionAtMostOne})$Out$6$

Known value · FindStat St001138

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