- FindStat

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Identifier

St001276: Dyck paths ⟶ ℤ

Values

[1,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

[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] => 0

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

>>> Load all 625 entries. <<<

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Description

The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by OEIS:A035929 .

Code

DeclareOperation("exttest",[IsList]);

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

local A,M,g,RegA,t,U;

A:=LIST[1];
M:=LIST[2];
g:=GorensteinDimensionOfAlgebra(A,30);
RegA:=DirectSumOfQPAModules(IndecProjectiveModules(A));
t:=Size(HomOverAlgebra(M,RegA));
U:=[];for i in [0..g-1] do Append(U,[Size(ExtOverAlgebra(NthSyzygy(M,i),RegA)[2])]);od;
return(t+Sum(U));
end);



DeclareOperation("generalkregular",[IsList]);

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

local A,M,g,RegA,t,U,k,L,W,WW;

A:=LIST[1];
k:=LIST[2];
L:=ARQuiverNak([A]);
RegA:=DirectSumOfQPAModules(IndecProjectiveModules(A));
W:=Filtered(L,x->ProjDimensionOfModule(x,30)=k and Size(HomOverAlgebra(x,RegA))=0);
WW:=Filtered(W,x->exttest([A,x])=1);
return(Size(WW));
end);

Created

Oct 18, 2018 at 20:27 by Rene Marczinzik

Updated

Oct 19, 2018 at 14:32 by Rene Marczinzik