- FindStat

Identifier

St001278: Dyck paths ⟶ ℤ

Values

[1,0] => 0

[1,0,1,0] => 1

[1,1,0,0] => 0

[1,0,1,0,1,0] => 2

[1,0,1,1,0,0] => 1

[1,1,0,0,1,0] => 1

[1,1,0,1,0,0] => 2

[1,1,1,0,0,0] => 0

[1,0,1,0,1,0,1,0] => 3

[1,0,1,0,1,1,0,0] => 2

[1,0,1,1,0,0,1,0] => 2

[1,0,1,1,0,1,0,0] => 3

[1,0,1,1,1,0,0,0] => 1

[1,1,0,0,1,0,1,0] => 2

[1,1,0,0,1,1,0,0] => 1

[1,1,0,1,0,0,1,0] => 3

[1,1,0,1,0,1,0,0] => 4

[1,1,0,1,1,0,0,0] => 2

[1,1,1,0,0,0,1,0] => 1

[1,1,1,0,0,1,0,0] => 2

[1,1,1,0,1,0,0,0] => 3

[1,1,1,1,0,0,0,0] => 0

[1,0,1,0,1,0,1,0,1,0] => 4

[1,0,1,0,1,0,1,1,0,0] => 3

[1,0,1,0,1,1,0,0,1,0] => 3

[1,0,1,0,1,1,0,1,0,0] => 4

[1,0,1,0,1,1,1,0,0,0] => 2

[1,0,1,1,0,0,1,0,1,0] => 3

[1,0,1,1,0,0,1,1,0,0] => 2

[1,0,1,1,0,1,0,0,1,0] => 4

[1,0,1,1,0,1,0,1,0,0] => 5

[1,0,1,1,0,1,1,0,0,0] => 3

[1,0,1,1,1,0,0,0,1,0] => 2

[1,0,1,1,1,0,0,1,0,0] => 3

[1,0,1,1,1,0,1,0,0,0] => 4

[1,0,1,1,1,1,0,0,0,0] => 1

[1,1,0,0,1,0,1,0,1,0] => 3

[1,1,0,0,1,0,1,1,0,0] => 2

[1,1,0,0,1,1,0,0,1,0] => 2

[1,1,0,0,1,1,0,1,0,0] => 3

[1,1,0,0,1,1,1,0,0,0] => 1

[1,1,0,1,0,0,1,0,1,0] => 4

[1,1,0,1,0,0,1,1,0,0] => 3

[1,1,0,1,0,1,0,0,1,0] => 5

[1,1,0,1,0,1,0,1,0,0] => 6

[1,1,0,1,0,1,1,0,0,0] => 4

[1,1,0,1,1,0,0,0,1,0] => 3

[1,1,0,1,1,0,0,1,0,0] => 4

[1,1,0,1,1,0,1,0,0,0] => 5

[1,1,0,1,1,1,0,0,0,0] => 2

[1,1,1,0,0,0,1,0,1,0] => 2

[1,1,1,0,0,0,1,1,0,0] => 1

[1,1,1,0,0,1,0,0,1,0] => 3

[1,1,1,0,0,1,0,1,0,0] => 4

[1,1,1,0,0,1,1,0,0,0] => 2

[1,1,1,0,1,0,0,0,1,0] => 4

[1,1,1,0,1,0,0,1,0,0] => 5

[1,1,1,0,1,0,1,0,0,0] => 6

[1,1,1,0,1,1,0,0,0,0] => 3

[1,1,1,1,0,0,0,0,1,0] => 1

[1,1,1,1,0,0,0,1,0,0] => 2

[1,1,1,1,0,0,1,0,0,0] => 3

[1,1,1,1,0,1,0,0,0,0] => 4

[1,1,1,1,1,0,0,0,0,0] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => 5

[1,0,1,0,1,0,1,0,1,1,0,0] => 4

[1,0,1,0,1,0,1,1,0,0,1,0] => 4

[1,0,1,0,1,0,1,1,0,1,0,0] => 5

[1,0,1,0,1,0,1,1,1,0,0,0] => 3

[1,0,1,0,1,1,0,0,1,0,1,0] => 4

[1,0,1,0,1,1,0,0,1,1,0,0] => 3

[1,0,1,0,1,1,0,1,0,0,1,0] => 5

[1,0,1,0,1,1,0,1,0,1,0,0] => 6

[1,0,1,0,1,1,0,1,1,0,0,0] => 4

[1,0,1,0,1,1,1,0,0,0,1,0] => 3

[1,0,1,0,1,1,1,0,0,1,0,0] => 4

[1,0,1,0,1,1,1,0,1,0,0,0] => 5

[1,0,1,0,1,1,1,1,0,0,0,0] => 2

[1,0,1,1,0,0,1,0,1,0,1,0] => 4

[1,0,1,1,0,0,1,0,1,1,0,0] => 3

[1,0,1,1,0,0,1,1,0,0,1,0] => 3

[1,0,1,1,0,0,1,1,0,1,0,0] => 4

[1,0,1,1,0,0,1,1,1,0,0,0] => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => 5

[1,0,1,1,0,1,0,0,1,1,0,0] => 4

[1,0,1,1,0,1,0,1,0,0,1,0] => 6

[1,0,1,1,0,1,0,1,0,1,0,0] => 7

[1,0,1,1,0,1,0,1,1,0,0,0] => 5

[1,0,1,1,0,1,1,0,0,0,1,0] => 4

[1,0,1,1,0,1,1,0,0,1,0,0] => 5

[1,0,1,1,0,1,1,0,1,0,0,0] => 6

[1,0,1,1,0,1,1,1,0,0,0,0] => 3

[1,0,1,1,1,0,0,0,1,0,1,0] => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => 4

[1,0,1,1,1,0,0,1,0,1,0,0] => 5

[1,0,1,1,1,0,0,1,1,0,0,0] => 3

[1,0,1,1,1,0,1,0,0,0,1,0] => 5

[1,0,1,1,1,0,1,0,0,1,0,0] => 6

[1,0,1,1,1,0,1,0,1,0,0,0] => 7

[1,0,1,1,1,0,1,1,0,0,0,0] => 4

>>> Load all 196 entries. <<<

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Description

The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.

References

[1] Iyama, O., Solberg, Øyvind Auslander-Gorenstein algebras and precluster tilting. zbMATH:06833443

Code

DeclareOperation("highertaufixall",[IsList]);

InstallMethod(highertaufixall, "for a representation of a quiver", [IsList],0,function(LIST)

local A,RegA,CoRegA,t,simA,U,L;

A:=LIST[1];
t:=LIST[2];
L:=ARQuiverNak([A]);
U:=Filtered(L,x->  IsomorphicModules(highertauinverse([DTr(NthSyzygy(x,t)),t]),x)=true);
return(Size(U));

end);

Created

Oct 20, 2018 at 20:09 by Rene Marczinzik

Updated

Nov 26, 2018 at 21:48 by Rene Marczinzik