NakayamaDominantDimension

The dominant dimension of the path's linear Nakayama algebra.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000685via DyckPath✓↗

NakayamaDominantDimension(path)The dominant dimension of the path's linear Nakayama algebra.

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 St000685. An algebra and its opposite have the same dominant dimension, which for the opposite is the least codominant dimension of a non-projective indecomposable injective module.

Definition

Epsil, as the implementation runs it.

on DyckPath, over _x

0 if Length(_x) < 1 else Apply(
                                (series: list<integer>) => Apply((leftSeries: list<integer>) => Apply(
                                                                                                       (dims: list<integer>) => Apply(
                                                                                                                                       (injectiveDims: list<integer>) => Apply(
                                                                                                                                                                                (codominantDims: list<integer>) => Min(Map(
                                                                                                                                                                                                                            Function(codominantDims[v], v),
                                                                                                                                                                                                                            Filter(Range(1, Length(_x) / 2 +
                                                                                                                                                                                                                                            1), Function(injectiveDims[v] > 0, v)),
                                                                                                                                                                                                                          )),
                                                                                                                                                                                Map(Function(Apply(
                                                                                                                                                                                                    (top: integer) => Fold(
                                                                                                                                                                                                                            Function(Apply((entry: integer) => Apply(
                                                                                                                                                                                                                                                                      (flag: integer) => [a[1], a[2], a[3] + flag, flag] if a[2] == entry else [a[1] + a[2], entry - a[2], a[3] + flag, flag],
                                                                                                                                                                                                                                                                      a[4] *
                                                                                                                                                                                                                                                                      (1 if leftSeries[a[1] + 1] <=
                                                                                                                                                                                                                                                                            entry else 0),
                                                                                                                                                                                                                                                                    ), series[a[1] + 1]), a, b),
                                                                                                                                                                                                                            [top, v - top, 0, 1],
                                                                                                                                                                                                                            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)),
                                                                                                                                                                              ),
                                                                                                                                       Map(Function(Apply(
                                                                                                                                                           (top: 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),
                                                                                                                                                                                   [top, v - top, 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)),
                                                                                                                                     ),
                                                                                                       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),
                                                                                                          ),
                                                                                                     ), Join([0], series)),
                                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$\mathrm{NakayamaDominantDimension}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St000685

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

live ↗

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

Known value · FindStat St000685

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

live ↗

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

Known value · FindStat St000685

About

Examples	5
Crosswalk	FindStat