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 of the linear Nakayama algebra corresponding to a Dyck path, minus the number of such modules that have reflexive Auslander–Reiten sequences.
Here an indecomposable non-injective projective module $P$ is said to have reflexive Auslander–Reiten sequences if every term in the Auslander–Reiten sequence for $P$ is reflexive.
The Dyck paths where the statistic returns the value zero are of special interest, see [1].
The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
Here an indecomposable non-injective projective module $P$ is said to have reflexive Auslander–Reiten sequences if every term in the Auslander–Reiten sequence for $P$ is reflexive.
The Dyck paths where the statistic returns the value zero are of special interest, see [1].
The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
References
[1] , Tachikawa, H. Reflexive Auslander-Reiten sequences MathSciNet:1048418 zbMATH:0686.16023
Code
gap('LoadPackage("QPA");')
import tempfile as _tf, os as _os
_gap_code = r"""
DeclareOperation("IsReflexive", [IsList]);
InstallMethod(IsReflexive, "for a representation of a quiver", [IsList],0,function(L)
local A, CoRegA, SS, 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 A, LL, i, prnotinjA, projA, tr, tulu;
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
);
"""
with _tf.NamedTemporaryFile(mode="w", suffix=".g", delete=False, dir="/tmp") as _f:
_f.write('LoadPackage("QPA");;\n')
_f.write(_gap_code)
_tmp = _f.name
gap.eval('Read("' + _tmp + '");')
_os.unlink(_tmp)
def kupisch(D):
DR = D.reverse()
H = DR.heights()
return [1 + H[i] for i, s in enumerate(DR) if s == 0] + [1]
def statistic(D):
K = kupisch(D)
return ZZ(gap.NumberreflexiveARseq2([K]))
Created
Nov 26, 2018 at 22:12 by Rene Marczinzik
Updated
Mar 13, 2026 at 15:02 by Nupur Jain
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