RegularModuleSecondInjectiveTermProjectiveDimension

The greatest projective dimension of the second term of the minimal injective coresolution of an indecomposable projective module.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001273via DyckPath✓↗

RegularModuleSecondInjectiveTermProjectiveDimension(path)The greatest projective dimension of the second term of the minimal injective coresolution of an indecomposable projective 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 St001273, which calls this the first term: its rows count the injective hull as the zeroth term, the injective hull of the first cosyzygy as the first.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

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

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001273

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

live ↗

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

Known value · FindStat St001273

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

live ↗

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

Known value · FindStat St001273

About

Examples	5
Crosswalk	FindStat