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Identifier
Values
=>
Cc0005;cc-rep
[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]=>1 [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]=>1 [1,0,1,1,0,1,0,0]=>2 [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]=>1 [1,1,0,1,0,1,0,0]=>2 [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]=>4 [1,0,1,0,1,0,1,1,0,0]=>3 [1,0,1,0,1,1,0,0,1,0]=>2 [1,0,1,0,1,1,0,1,0,0]=>3 [1,0,1,0,1,1,1,0,0,0]=>2 [1,0,1,1,0,0,1,0,1,0]=>2 [1,0,1,1,0,0,1,1,0,0]=>1 [1,0,1,1,0,1,0,0,1,0]=>2 [1,0,1,1,0,1,0,1,0,0]=>3 [1,0,1,1,0,1,1,0,0,0]=>2 [1,0,1,1,1,0,0,0,1,0]=>1 [1,0,1,1,1,0,0,1,0,0]=>1 [1,0,1,1,1,0,1,0,0,0]=>2 [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]=>1 [1,1,0,0,1,1,0,1,0,0]=>2 [1,1,0,0,1,1,1,0,0,0]=>1 [1,1,0,1,0,0,1,0,1,0]=>2 [1,1,0,1,0,0,1,1,0,0]=>1 [1,1,0,1,0,1,0,0,1,0]=>3 [1,1,0,1,0,1,0,1,0,0]=>2 [1,1,0,1,0,1,1,0,0,0]=>2 [1,1,0,1,1,0,0,0,1,0]=>1 [1,1,0,1,1,0,0,1,0,0]=>1 [1,1,0,1,1,0,1,0,0,0]=>2 [1,1,0,1,1,1,0,0,0,0]=>1 [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]=>1 [1,1,1,0,0,1,0,1,0,0]=>2 [1,1,1,0,0,1,1,0,0,0]=>1 [1,1,1,0,1,0,0,0,1,0]=>1 [1,1,1,0,1,0,0,1,0,0]=>1 [1,1,1,0,1,0,1,0,0,0]=>2 [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]=>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]=>3 [1,0,1,0,1,0,1,1,0,1,0,0]=>4 [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]=>2 [1,0,1,0,1,1,0,1,0,0,1,0]=>3 [1,0,1,0,1,1,0,1,0,1,0,0]=>4 [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]=>2 [1,0,1,0,1,1,1,0,0,1,0,0]=>2 [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]=>2 [1,0,1,1,0,0,1,0,1,0,1,0]=>3 [1,0,1,1,0,0,1,0,1,1,0,0]=>2 [1,0,1,1,0,0,1,1,0,0,1,0]=>1 [1,0,1,1,0,0,1,1,0,1,0,0]=>2 [1,0,1,1,0,0,1,1,1,0,0,0]=>1 [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]=>2 [1,0,1,1,0,1,0,1,0,0,1,0]=>4 [1,0,1,1,0,1,0,1,0,1,0,0]=>3 [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]=>2 [1,0,1,1,0,1,1,0,0,1,0,0]=>2 [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]=>2 [1,0,1,1,1,0,0,0,1,0,1,0]=>2 [1,0,1,1,1,0,0,0,1,1,0,0]=>1 [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]=>1 [1,0,1,1,1,0,1,0,0,0,1,0]=>2 [1,0,1,1,1,0,1,0,0,1,0,0]=>2 [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]=>2 [1,0,1,1,1,1,0,0,0,0,1,0]=>1 [1,0,1,1,1,1,0,0,0,1,0,0]=>1 [1,0,1,1,1,1,0,0,1,0,0,0]=>1 [1,0,1,1,1,1,0,1,0,0,0,0]=>2 [1,0,1,1,1,1,1,0,0,0,0,0]=>1 [1,1,0,0,1,0,1,0,1,0,1,0]=>4 [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]=>2 [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]=>2 [1,1,0,0,1,1,0,0,1,0,1,0]=>2 [1,1,0,0,1,1,0,0,1,1,0,0]=>1 [1,1,0,0,1,1,0,1,0,0,1,0]=>2 [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]=>2 [1,1,0,0,1,1,1,0,0,0,1,0]=>1 [1,1,0,0,1,1,1,0,0,1,0,0]=>1 [1,1,0,0,1,1,1,0,1,0,0,0]=>2 [1,1,0,0,1,1,1,1,0,0,0,0]=>1 [1,1,0,1,0,0,1,0,1,0,1,0]=>3 [1,1,0,1,0,0,1,0,1,1,0,0]=>2 [1,1,0,1,0,0,1,1,0,0,1,0]=>1 [1,1,0,1,0,0,1,1,0,1,0,0]=>2 [1,1,0,1,0,0,1,1,1,0,0,0]=>1 [1,1,0,1,0,1,0,0,1,0,1,0]=>4 [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]=>3 [1,1,0,1,0,1,0,1,0,1,0,0]=>3 [1,1,0,1,0,1,0,1,1,0,0,0]=>2 [1,1,0,1,0,1,1,0,0,0,1,0]=>2 [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]=>2 [1,1,0,1,0,1,1,1,0,0,0,0]=>2 [1,1,0,1,1,0,0,0,1,0,1,0]=>2 [1,1,0,1,1,0,0,0,1,1,0,0]=>1 [1,1,0,1,1,0,0,1,0,0,1,0]=>1 [1,1,0,1,1,0,0,1,0,1,0,0]=>2 [1,1,0,1,1,0,0,1,1,0,0,0]=>1 [1,1,0,1,1,0,1,0,0,0,1,0]=>2 [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]=>2 [1,1,0,1,1,0,1,1,0,0,0,0]=>2 [1,1,0,1,1,1,0,0,0,0,1,0]=>1 [1,1,0,1,1,1,0,0,0,1,0,0]=>1 [1,1,0,1,1,1,0,0,1,0,0,0]=>1 [1,1,0,1,1,1,0,1,0,0,0,0]=>2 [1,1,0,1,1,1,1,0,0,0,0,0]=>1 [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]=>2 [1,1,1,0,0,0,1,1,0,0,1,0]=>1 [1,1,1,0,0,0,1,1,0,1,0,0]=>2 [1,1,1,0,0,0,1,1,1,0,0,0]=>1 [1,1,1,0,0,1,0,0,1,0,1,0]=>2 [1,1,1,0,0,1,0,0,1,1,0,0]=>1 [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]=>2 [1,1,1,0,0,1,0,1,1,0,0,0]=>2 [1,1,1,0,0,1,1,0,0,0,1,0]=>1 [1,1,1,0,0,1,1,0,0,1,0,0]=>1 [1,1,1,0,0,1,1,0,1,0,0,0]=>2 [1,1,1,0,0,1,1,1,0,0,0,0]=>1 [1,1,1,0,1,0,0,0,1,0,1,0]=>2 [1,1,1,0,1,0,0,0,1,1,0,0]=>1 [1,1,1,0,1,0,0,1,0,0,1,0]=>1 [1,1,1,0,1,0,0,1,0,1,0,0]=>2 [1,1,1,0,1,0,0,1,1,0,0,0]=>1 [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]=>2 [1,1,1,0,1,0,1,0,1,0,0,0]=>2 [1,1,1,0,1,0,1,1,0,0,0,0]=>2 [1,1,1,0,1,1,0,0,0,0,1,0]=>1 [1,1,1,0,1,1,0,0,0,1,0,0]=>1 [1,1,1,0,1,1,0,0,1,0,0,0]=>1 [1,1,1,0,1,1,0,1,0,0,0,0]=>2 [1,1,1,0,1,1,1,0,0,0,0,0]=>1 [1,1,1,1,0,0,0,0,1,0,1,0]=>2 [1,1,1,1,0,0,0,0,1,1,0,0]=>1 [1,1,1,1,0,0,0,1,0,0,1,0]=>1 [1,1,1,1,0,0,0,1,0,1,0,0]=>2 [1,1,1,1,0,0,0,1,1,0,0,0]=>1 [1,1,1,1,0,0,1,0,0,0,1,0]=>1 [1,1,1,1,0,0,1,0,0,1,0,0]=>1 [1,1,1,1,0,0,1,0,1,0,0,0]=>2 [1,1,1,1,0,0,1,1,0,0,0,0]=>1 [1,1,1,1,0,1,0,0,0,0,1,0]=>1 [1,1,1,1,0,1,0,0,0,1,0,0]=>1 [1,1,1,1,0,1,0,0,1,0,0,0]=>1 [1,1,1,1,0,1,0,1,0,0,0,0]=>2 [1,1,1,1,0,1,1,0,0,0,0,0]=>1 [1,1,1,1,1,0,0,0,0,0,1,0]=>1 [1,1,1,1,1,0,0,0,0,1,0,0]=>1 [1,1,1,1,1,0,0,0,1,0,0,0]=>1 [1,1,1,1,1,0,0,1,0,0,0,0]=>1 [1,1,1,1,1,0,1,0,0,0,0,0]=>1 [1,1,1,1,1,1,0,0,0,0,0,0]=>0
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Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras.
The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers oeis.org/A001006.
Code
DeclareOperation("IsNtorsionfree",[IsList]);

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

local A,M,n,CoRegA,temm23;

A:=LIST[1];
M:=LIST[2];

n:=LIST[3];

CoRegA:=DirectSumOfQPAModules(IndecInjectiveModules(A));
temm23:=[];
for i in [0..n-1] do Append(temm23,[Size(ExtOverAlgebra(NthSyzygy(CoRegA,i),DTr(M))[2])]);od;
return(Sum(temm23));
end);



DeclareOperation("torsionfreeindex",[IsList]);

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

local A,M,n,CoRegA,temm23,U,g;

A:=LIST[1];
M:=LIST[2];
g:=LIST[3];
U:=Filtered([1..g],x->IsNtorsionfree([A,M,x])>0);
return(Minimum(U)-1);
end);




DeclareOperation("torsionmax",[IsList]);

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

local A,M,n,CoRegA,temm23,simA,UU,g;

A:=LIST[1];
g:=GlobalDimensionOfAlgebra(A,30);
simA:=Filtered(SimpleModules(A),x->IsProjectiveModule(x)=false);
UU:=[];for i in simA do Append(UU,[torsionfreeindex([A,i,g])]);od;
return(Maximum(UU));
end);



Created
Nov 24, 2018 at 16:21 by Rene Marczinzik
Updated
Nov 24, 2018 at 16:21 by Rene Marczinzik