Identifier
Identifier
• St001187: ⟶ ℤ (values match St000024The number of double up and double down steps of a Dyck path., St000443The number of long tunnels of a Dyck path., St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path., St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.)
Values
[1,0] => 1
[1,0,1,0] => 1
[1,1,0,0] => 2
[1,0,1,0,1,0] => 1
[1,0,1,1,0,0] => 2
[1,1,0,0,1,0] => 2
[1,1,0,1,0,0] => 2
[1,1,1,0,0,0] => 3
[1,0,1,0,1,0,1,0] => 1
[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] => 2
[1,0,1,1,1,0,0,0] => 3
[1,1,0,0,1,0,1,0] => 2
[1,1,0,0,1,1,0,0] => 3
[1,1,0,1,0,0,1,0] => 2
[1,1,0,1,0,1,0,0] => 2
[1,1,0,1,1,0,0,0] => 3
[1,1,1,0,0,0,1,0] => 3
[1,1,1,0,0,1,0,0] => 3
[1,1,1,0,1,0,0,0] => 3
[1,1,1,1,0,0,0,0] => 4
[1,0,1,0,1,0,1,0,1,0] => 1
[1,0,1,0,1,0,1,1,0,0] => 2
[1,0,1,0,1,1,0,0,1,0] => 2
[1,0,1,0,1,1,0,1,0,0] => 2
[1,0,1,0,1,1,1,0,0,0] => 3
[1,0,1,1,0,0,1,0,1,0] => 2
[1,0,1,1,0,0,1,1,0,0] => 3
[1,0,1,1,0,1,0,0,1,0] => 2
[1,0,1,1,0,1,0,1,0,0] => 2
[1,0,1,1,0,1,1,0,0,0] => 3
[1,0,1,1,1,0,0,0,1,0] => 3
[1,0,1,1,1,0,0,1,0,0] => 3
[1,0,1,1,1,0,1,0,0,0] => 3
[1,0,1,1,1,1,0,0,0,0] => 4
[1,1,0,0,1,0,1,0,1,0] => 2
[1,1,0,0,1,0,1,1,0,0] => 3
[1,1,0,0,1,1,0,0,1,0] => 3
[1,1,0,0,1,1,0,1,0,0] => 3
[1,1,0,0,1,1,1,0,0,0] => 4
[1,1,0,1,0,0,1,0,1,0] => 2
[1,1,0,1,0,0,1,1,0,0] => 3
[1,1,0,1,0,1,0,0,1,0] => 2
[1,1,0,1,0,1,0,1,0,0] => 2
[1,1,0,1,0,1,1,0,0,0] => 3
[1,1,0,1,1,0,0,0,1,0] => 3
[1,1,0,1,1,0,0,1,0,0] => 3
[1,1,0,1,1,0,1,0,0,0] => 3
[1,1,0,1,1,1,0,0,0,0] => 4
[1,1,1,0,0,0,1,0,1,0] => 3
[1,1,1,0,0,0,1,1,0,0] => 4
[1,1,1,0,0,1,0,0,1,0] => 3
[1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,0,0,1,1,0,0,0] => 4
[1,1,1,0,1,0,0,0,1,0] => 3
[1,1,1,0,1,0,0,1,0,0] => 3
[1,1,1,0,1,0,1,0,0,0] => 3
[1,1,1,0,1,1,0,0,0,0] => 4
[1,1,1,1,0,0,0,0,1,0] => 4
[1,1,1,1,0,0,0,1,0,0] => 4
[1,1,1,1,0,0,1,0,0,0] => 4
[1,1,1,1,0,1,0,0,0,0] => 4
[1,1,1,1,1,0,0,0,0,0] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => 2
[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] => 2
[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] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => 3
[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] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => 2
[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] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => 4
[1,0,1,1,0,1,0,0,1,0,1,0] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => 4
[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] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => 4
[1,0,1,1,1,1,0,0,0,0,1,0] => 4
[1,0,1,1,1,1,0,0,0,1,0,0] => 4
[1,0,1,1,1,1,0,0,1,0,0,0] => 4
[1,0,1,1,1,1,0,1,0,0,0,0] => 4
[1,0,1,1,1,1,1,0,0,0,0,0] => 5
[1,1,0,0,1,0,1,0,1,0,1,0] => 2
[1,1,0,0,1,0,1,0,1,1,0,0] => 3
[1,1,0,0,1,0,1,1,0,0,1,0] => 3
[1,1,0,0,1,0,1,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,1,1,0,0,0] => 4
[1,1,0,0,1,1,0,0,1,0,1,0] => 3
[1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,0,1,1,0,1,0,0,1,0] => 3
[1,1,0,0,1,1,0,1,0,1,0,0] => 3
[1,1,0,0,1,1,0,1,1,0,0,0] => 4
[1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => 4
[1,1,0,0,1,1,1,0,1,0,0,0] => 4
[1,1,0,0,1,1,1,1,0,0,0,0] => 5
[1,1,0,1,0,0,1,0,1,0,1,0] => 2
[1,1,0,1,0,0,1,0,1,1,0,0] => 3
[1,1,0,1,0,0,1,1,0,0,1,0] => 3
[1,1,0,1,0,0,1,1,0,1,0,0] => 3
[1,1,0,1,0,0,1,1,1,0,0,0] => 4
[1,1,0,1,0,1,0,0,1,0,1,0] => 2
[1,1,0,1,0,1,0,0,1,1,0,0] => 3
[1,1,0,1,0,1,0,1,0,0,1,0] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => 2
[1,1,0,1,0,1,0,1,1,0,0,0] => 3
[1,1,0,1,0,1,1,0,0,0,1,0] => 3
[1,1,0,1,0,1,1,0,0,1,0,0] => 3
[1,1,0,1,0,1,1,0,1,0,0,0] => 3
[1,1,0,1,0,1,1,1,0,0,0,0] => 4
[1,1,0,1,1,0,0,0,1,0,1,0] => 3
[1,1,0,1,1,0,0,0,1,1,0,0] => 4
[1,1,0,1,1,0,0,1,0,0,1,0] => 3
[1,1,0,1,1,0,0,1,0,1,0,0] => 3
[1,1,0,1,1,0,0,1,1,0,0,0] => 4
[1,1,0,1,1,0,1,0,0,0,1,0] => 3
[1,1,0,1,1,0,1,0,0,1,0,0] => 3
[1,1,0,1,1,0,1,0,1,0,0,0] => 3
[1,1,0,1,1,0,1,1,0,0,0,0] => 4
[1,1,0,1,1,1,0,0,0,0,1,0] => 4
[1,1,0,1,1,1,0,0,0,1,0,0] => 4
[1,1,0,1,1,1,0,0,1,0,0,0] => 4
[1,1,0,1,1,1,0,1,0,0,0,0] => 4
[1,1,0,1,1,1,1,0,0,0,0,0] => 5
[1,1,1,0,0,0,1,0,1,0,1,0] => 3
[1,1,1,0,0,0,1,0,1,1,0,0] => 4
[1,1,1,0,0,0,1,1,0,0,1,0] => 4
[1,1,1,0,0,0,1,1,0,1,0,0] => 4
[1,1,1,0,0,0,1,1,1,0,0,0] => 5
[1,1,1,0,0,1,0,0,1,0,1,0] => 3
[1,1,1,0,0,1,0,0,1,1,0,0] => 4
[1,1,1,0,0,1,0,1,0,0,1,0] => 3
[1,1,1,0,0,1,0,1,0,1,0,0] => 3
[1,1,1,0,0,1,0,1,1,0,0,0] => 4
[1,1,1,0,0,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,1,1,0,0,1,0,0] => 4
[1,1,1,0,0,1,1,0,1,0,0,0] => 4
[1,1,1,0,0,1,1,1,0,0,0,0] => 5
[1,1,1,0,1,0,0,0,1,0,1,0] => 3
[1,1,1,0,1,0,0,0,1,1,0,0] => 4
[1,1,1,0,1,0,0,1,0,0,1,0] => 3
[1,1,1,0,1,0,0,1,0,1,0,0] => 3
[1,1,1,0,1,0,0,1,1,0,0,0] => 4
[1,1,1,0,1,0,1,0,0,0,1,0] => 3
[1,1,1,0,1,0,1,0,0,1,0,0] => 3
[1,1,1,0,1,0,1,0,1,0,0,0] => 3
[1,1,1,0,1,0,1,1,0,0,0,0] => 4
[1,1,1,0,1,1,0,0,0,0,1,0] => 4
[1,1,1,0,1,1,0,0,0,1,0,0] => 4
[1,1,1,0,1,1,0,0,1,0,0,0] => 4
[1,1,1,0,1,1,0,1,0,0,0,0] => 4
[1,1,1,0,1,1,1,0,0,0,0,0] => 5
[1,1,1,1,0,0,0,0,1,0,1,0] => 4
[1,1,1,1,0,0,0,0,1,1,0,0] => 5
[1,1,1,1,0,0,0,1,0,0,1,0] => 4
[1,1,1,1,0,0,0,1,0,1,0,0] => 4
[1,1,1,1,0,0,0,1,1,0,0,0] => 5
[1,1,1,1,0,0,1,0,0,0,1,0] => 4
[1,1,1,1,0,0,1,0,0,1,0,0] => 4
[1,1,1,1,0,0,1,0,1,0,0,0] => 4
[1,1,1,1,0,0,1,1,0,0,0,0] => 5
[1,1,1,1,0,1,0,0,0,0,1,0] => 4
[1,1,1,1,0,1,0,0,0,1,0,0] => 4
[1,1,1,1,0,1,0,0,1,0,0,0] => 4
[1,1,1,1,0,1,0,1,0,0,0,0] => 4
[1,1,1,1,0,1,1,0,0,0,0,0] => 5
[1,1,1,1,1,0,0,0,0,0,1,0] => 5
[1,1,1,1,1,0,0,0,0,1,0,0] => 5
[1,1,1,1,1,0,0,0,1,0,0,0] => 5
[1,1,1,1,1,0,0,1,0,0,0,0] => 5
[1,1,1,1,1,0,1,0,0,0,0,0] => 5
[1,1,1,1,1,1,0,0,0,0,0,0] => 6
Description
The number of simple modules with grade at least one in the corresponding Nakayama algebra.
Code
DeclareOperation("gradeofmodule",[IsList]);

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

local A,M,RegA,g,temmi,UT;

A:=LIST[1];
M:=LIST[2];
RegA:=DirectSumOfQPAModules(IndecProjectiveModules(A));
g:=GorensteinDimensionOfAlgebra(A,30);
temmi:=[];Append(temmi,[Size(HomOverAlgebra(M,RegA))]);
for i in [0..g-1] do Append(temmi,[Size(ExtOverAlgebra(NthSyzygy(M,i),RegA)[2])]);od;
UT:=Filtered([0..g],x->temmi[x+1]>0);
return(Minimum(UT));
end);

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

local A,k,simA,WW;

A:=LIST[1];
k:=LIST[2];
simA:=SimpleModules(A);