CompositionNakayamaGlobalDimension

The global dimension of the linear Nakayama algebra whose Kupisch series is c_1 + 1, ..., 2, c_2 + 1, ..., 2, ..., 1.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001235via IntegerComposition✓↗

CompositionNakayamaGlobalDimension(composition)The global dimension of the linear Nakayama algebra whose Kupisch series is c_1 + 1, ..., 2, c_2 + 1, ..., 2, ..., 1.

Typed signatures

Type	Package	Applies
(composition) -> number	combinatorics	always

Domain: Composition statistics

Details

• Defined over IntegerComposition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerComposition 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 St001235 (its Comp-Nakayama algebra): the largest projective dimension of a simple module.

Definition

Epsil, as the implementation runs it.

on IntegerComposition, over _x

0 if Length(_x) < 1 else Apply(
                                (series: list<integer>) => Max(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, Sum(_x) + 1),
                                                                                 )[3], v),
                                                                    Range(
                                                                           1,
                                                                           Sum(_x) +
                                                                           1,
                                                                         ),
                                                                  )),
                                Join(
                                      Fold(Function(Join(
                                                          a,
                                                          Map(
                                                               Function(b + 2 -
                                                                        t, t),
                                                               Range(1, b),
                                                             ),
                                                        ), a, b), [], _x),
                                      [1],
                                    ),
                              )

Examples

Basic #

the composition $[2,1,3,1,1]$

live ↗

In$\mathrm{CompositionNakayamaGlobalDimension}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]))$Out$3$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([1]), \text{CompositionNakayamaGlobalDimension})$Out$1$

Known value · FindStat St001235

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([2]), \text{CompositionNakayamaGlobalDimension})$Out$1$

Known value · FindStat St001235

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([3]), \text{CompositionNakayamaGlobalDimension})$Out$1$

Known value · FindStat St001235

About

Examples	4
Crosswalk	FindStat