Identifier
- St001297: Dyck paths ⟶ ℤ
Values
=>
Cc0005;cc-rep
[1,0]=>1
[1,0,1,0]=>1
[1,1,0,0]=>2
[1,0,1,0,1,0]=>0
[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]=>0
[1,0,1,0,1,1,0,0]=>1
[1,0,1,1,0,0,1,0]=>2
[1,0,1,1,0,1,0,0]=>1
[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]=>1
[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]=>0
[1,0,1,0,1,0,1,1,0,0]=>1
[1,0,1,0,1,1,0,0,1,0]=>1
[1,0,1,0,1,1,0,1,0,0]=>1
[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]=>3
[1,0,1,1,0,1,0,0,1,0]=>1
[1,0,1,1,0,1,0,1,0,0]=>1
[1,0,1,1,0,1,1,0,0,0]=>2
[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]=>2
[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]=>1
[1,1,0,1,0,1,0,1,0,0]=>0
[1,1,0,1,0,1,1,0,0,0]=>2
[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]=>2
[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]=>2
[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]=>0
[1,0,1,0,1,0,1,0,1,1,0,0]=>1
[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]=>1
[1,0,1,0,1,0,1,1,1,0,0,0]=>2
[1,0,1,0,1,1,0,0,1,0,1,0]=>1
[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]=>1
[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]=>2
[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]=>2
[1,0,1,0,1,1,1,1,0,0,0,0]=>3
[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]=>1
[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]=>1
[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]=>2
[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]=>2
[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]=>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]=>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]=>2
[1,0,1,1,1,0,1,1,0,0,0,0]=>3
[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]=>3
[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]=>1
[1,1,0,1,0,1,0,0,1,1,0,0]=>2
[1,1,0,1,0,1,0,1,0,0,1,0]=>0
[1,1,0,1,0,1,0,1,0,1,0,0]=>0
[1,1,0,1,0,1,0,1,1,0,0,0]=>1
[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]=>2
[1,1,0,1,0,1,1,0,1,0,0,0]=>1
[1,1,0,1,0,1,1,1,0,0,0,0]=>3
[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]=>2
[1,1,0,1,1,0,1,0,0,1,0,0]=>2
[1,1,0,1,1,0,1,0,1,0,0,0]=>1
[1,1,0,1,1,0,1,1,0,0,0,0]=>3
[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]=>3
[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]=>2
[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]=>2
[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]=>1
[1,1,1,0,1,0,1,1,0,0,0,0]=>3
[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]=>3
[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]=>3
[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
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Description
The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
Here an indecomposable non-injective projective module P is said to have reflexive Auslander-Reiten sequences in case every term in the Auslander-Reiten sequence for P is reflexive.
The Dyck paths where the statistic returns the value 0 are of special interesting, see [1].
Here an indecomposable non-injective projective module P is said to have reflexive Auslander-Reiten sequences in case every term in the Auslander-Reiten sequence for P is reflexive.
The Dyck paths where the statistic returns the value 0 are of special interesting, see [1].
References
[1] , Tachikawa, H. Reflexive Auslander-Reiten sequences MathSciNet:1048418 zbMATH:0686.16023
Code
DeclareOperation("IsReflexive", [IsList]);
InstallMethod(IsReflexive, "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]);
dd2:=Size(ExtOverAlgebra(NthSyzygy(CoRegA,1),DTr(SS))[2]);
return(dd1+dd2);
end
);
DeclareOperation("HasProjreflexiveARseq", [IsList]);
InstallMethod(HasProjreflexiveARseq, "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(IsReflexive([A,UU1])+IsReflexive([A,UU2]));
end
);
DeclareOperation("NumberreflexiveARseq2", [IsList]);
InstallMethod(NumberreflexiveARseq2, "for a representation of a quiver", [IsList],0,function(L)
local LL,A,projA,prnotinjA,tulu,tr,i;
LL:=L[1];
A:=NakayamaAlgebra(LL,GF(3));
projA:=IndecProjectiveModules(A);prnotinjA:=Filtered(projA,x->IsInjectiveModule(x)=false);
tulu:=[];for i in prnotinjA do Append(tulu,[HasProjreflexiveARseq([A,i])]);od;
tr:=Filtered(tulu,x->(x=0));
return(Size(prnotinjA)-Size(tr));
end
);
Created
Nov 26, 2018 at 22:12 by Rene Marczinzik
Updated
Nov 26, 2018 at 22:12 by Rene Marczinzik
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