SimplesOfGradeAtLeastTwo

The simple modules of grade at least two.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001188via DyckPath✓↗St001244via DyckPath✓↗

SimplesOfGradeAtLeastTwo(path)The simple modules of grade at least two.

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 St001188, and St001244 (the simples of projective dimension one that are not 1-regular), which has the same values on every path of semilength 8 or less.

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(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),
                                                                                           [
                                                                                             v - 1,
                                                                                             1,
                                                                                             0,
                                                                                             1 if Max(Sum(Map(
                                                                                                               Function(Max(0, Min(1, w - 1 +
                                                                                                                                      series[w] -
                                                                                                                                      (v - 1), Min(1, series[w]) -
                                                                                                                                        (w - 1 +
                                                                                                                                         series[w] -
                                                                                                                                         (v - 1)) +
                                                                                                                                        1)), w),
                                                                                                               Range(1, Length(_x) / 2 +
                                                                                                                        1),
                                                                                                             )), 0) >
                                                                                                  0 else 0,
                                                                                             0,
                                                                                           ],
                                                                                           Range(1, Length(_x) / 2 +
                                                                                                    1),
                                                                                         )[3], v), Range(1, Length(_x) / 2 +
                                                                                                           1)), Function(d >= 2, 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$\mathrm{SimplesOfGradeAtLeastTwo}({\operatorname{DyckPath}}([1, 1, 0, 0, 1, 0]))$Out$1$

the step word $UDUUDD$

live ↗

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

Properties #

FindStat's value for [1,0]

live ↗

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

Known value · FindStat St001188

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

live ↗

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

Known value · FindStat St001188

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

live ↗

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

Known value · FindStat St001188

About

Examples	5
Crosswalk	FindStat ×2