- FindStat

Identifier

St001223: Dyck paths ⟶ ℤ (values match St000932The number of occurrences of the pattern UDU in a Dyck path., St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.)

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] => 0

[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] => 1

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

[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] => 0

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

[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] => 2

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

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

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

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

[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] => 2

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

[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] => 1

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

[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] => 0

[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] => 0

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

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

[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] => 3

[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] => 3

[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] => 3

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

[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

>>> Load all 196 entries. <<<

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Description

Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.

References

[1] Tachikawa, H. Reflexive Auslander-Reiten sequences. zbMATH:0686.16023

Code

DeclareOperation("IsTorsionfree", [IsList]);

InstallMethod(IsTorsionfree, "for a representation of a quiver", [IsList],0,function(L)

local A,SS,CoRegA,dd1,dd2;
A:=L[1];
SS:=L[2];
CoRegA:=DirectSumOfQPAModules(IndecInjectiveModules(A));
dd1:=Size(ExtOverAlgebra(CoRegA,DTr(SS))[2]);
return(dd1);
end
);
DeclareOperation("HasProjtorsionlessARseq", [IsList]);

InstallMethod(HasProjtorsionlessARseq, "for a representation of a quiver", [IsList],0,function(L)

local A,P,UU1,UU2;
A:=L[1];
P:=L[2];
UU1:=DTr(P,-1);
UU2:=Source(AlmostSplitSequence(UU1)[2]);
return(IsTorsionfree([A,UU1])+IsTorsionfree([A,UU2]));
end
);


DeclareOperation("NumbertorsionfreeARseq", [IsList]);

InstallMethod(NumbertorsionfreeARseq, "for a representation of a quiver", [IsList],0,function(L)

local A,simA,prnotinjA,tulu,tr;
L:=L[1];
A:=NakayamaAlgebra(L,GF(3));
simA:=SimpleModules(A);prnotinjA:=Filtered(simA,x->IsInjectiveModule(x)=false);
tulu:=[];for i in prnotinjA do Append(tulu,[HasProjtorsionlessARseq([A,i])]);od;
tr:=Filtered(tulu,x->(x=0));
return(Size(tr));
end
);

Created

Jul 11, 2018 at 13:58 by Rene Marczinzik

Updated

Jul 11, 2018 at 13:58 by Rene Marczinzik