NakayamaFinitisticDominantDimension

The greatest finite dominant dimension of an indecomposable module.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000686via DyckPath✓↗

NakayamaFinitisticDominantDimension(path)The greatest finite dominant dimension of an indecomposable 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 St000686. A dominant dimension reaching the vertex count is the infinite one the fold caps at.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (series: list<integer>) => Max(Map(
                                                                    Function(Max(Map(
                                                                                      Function(Apply((size: integer) => Apply(
                                                                                                                               (dominant: integer) => dominant if dominant <
                                                                                                                                                                  Length(_x) / 2 +
                                                                                                                                                                  1 else 0,
                                                                                                                               Fold(
                                                                                                                                     Function(Apply(
                                                                                                                                                     (socle: integer) => Apply(
                                                                                                                                                                                (hull: integer) => Apply(
                                                                                                                                                                                                          (flag: integer) => [a[1], 0, a[3] + flag, flag] if a[2] == 0 else [hull, a[1] - hull, a[3] + flag, flag],
                                                                                                                                                                                                          a[4] *
                                                                                                                                                                                                          (1 if a[2] == 0 ||
                                                                                                                                                                                                                series[hull + 1] ==
                                                                                                                                                                                                                socle + 1 -
                                                                                                                                                                                                                hull else 0),
                                                                                                                                                                                                        ),
                                                                                                                                                                                Min(Filter(
                                                                                                                                                                                            Range(0, socle, 1),
                                                                                                                                                                                            Function(x +
                                                                                                                                                                                                     series[x + 1] >
                                                                                                                                                                                                     socle, x),
                                                                                                                                                                                          )),
                                                                                                                                                                              ),
                                                                                                                                                     Max(a[1] + a[2] -
                                                                                                                                                         1, 0),
                                                                                                                                                   ), a, b),
                                                                                                                                     [v - 1, size, 0, 1],
                                                                                                                                     Range(1, Length(_x) / 2 +
                                                                                                                                              1),
                                                                                                                                   )[3],
                                                                                                                             ), y), y),
                                                                                      Range(1, Max(series[v], 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$\mathrm{NakayamaFinitisticDominantDimension}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$2$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000686

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

live ↗

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

Known value · FindStat St000686

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

live ↗

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

Known value · FindStat St000686

About

Examples	5
Crosswalk	FindStat