Identifier
- St001295: Dyck paths ⟶ ℤ (values match St000012The area of a Dyck path.)
Values
=>
Cc0005;cc-rep
[1,0]=>0
[1,0,1,0]=>0
[1,1,0,0]=>1
[1,0,1,0,1,0]=>0
[1,0,1,1,0,0]=>1
[1,1,0,0,1,0]=>1
[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]=>1
[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]=>1
[1,1,0,0,1,1,0,0]=>2
[1,1,0,1,0,0,1,0]=>2
[1,1,0,1,0,1,0,0]=>3
[1,1,0,1,1,0,0,0]=>4
[1,1,1,0,0,0,1,0]=>3
[1,1,1,0,0,1,0,0]=>4
[1,1,1,0,1,0,0,0]=>5
[1,1,1,1,0,0,0,0]=>6
[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]=>2
[1,0,1,0,1,1,1,0,0,0]=>3
[1,0,1,1,0,0,1,0,1,0]=>1
[1,0,1,1,0,0,1,1,0,0]=>2
[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]=>4
[1,0,1,1,1,0,0,0,1,0]=>3
[1,0,1,1,1,0,0,1,0,0]=>4
[1,0,1,1,1,0,1,0,0,0]=>5
[1,0,1,1,1,1,0,0,0,0]=>6
[1,1,0,0,1,0,1,0,1,0]=>1
[1,1,0,0,1,0,1,1,0,0]=>2
[1,1,0,0,1,1,0,0,1,0]=>2
[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]=>3
[1,1,0,1,0,1,0,1,0,0]=>4
[1,1,0,1,0,1,1,0,0,0]=>5
[1,1,0,1,1,0,0,0,1,0]=>4
[1,1,0,1,1,0,0,1,0,0]=>5
[1,1,0,1,1,0,1,0,0,0]=>6
[1,1,0,1,1,1,0,0,0,0]=>7
[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]=>4
[1,1,1,0,0,1,0,1,0,0]=>5
[1,1,1,0,0,1,1,0,0,0]=>6
[1,1,1,0,1,0,0,0,1,0]=>5
[1,1,1,0,1,0,0,1,0,0]=>6
[1,1,1,0,1,0,1,0,0,0]=>7
[1,1,1,0,1,1,0,0,0,0]=>8
[1,1,1,1,0,0,0,0,1,0]=>6
[1,1,1,1,0,0,0,1,0,0]=>7
[1,1,1,1,0,0,1,0,0,0]=>8
[1,1,1,1,0,1,0,0,0,0]=>9
[1,1,1,1,1,0,0,0,0,0]=>10
[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]=>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]=>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]=>2
[1,0,1,0,1,1,0,1,0,1,0,0]=>3
[1,0,1,0,1,1,0,1,1,0,0,0]=>4
[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]=>4
[1,0,1,0,1,1,1,0,1,0,0,0]=>5
[1,0,1,0,1,1,1,1,0,0,0,0]=>6
[1,0,1,1,0,0,1,0,1,0,1,0]=>1
[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]=>2
[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]=>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]=>5
[1,0,1,1,0,1,1,0,0,0,1,0]=>4
[1,0,1,1,0,1,1,0,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,1,0,0,0]=>6
[1,0,1,1,0,1,1,1,0,0,0,0]=>7
[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]=>4
[1,0,1,1,1,0,0,1,0,1,0,0]=>5
[1,0,1,1,1,0,0,1,1,0,0,0]=>6
[1,0,1,1,1,0,1,0,0,0,1,0]=>5
[1,0,1,1,1,0,1,0,0,1,0,0]=>6
[1,0,1,1,1,0,1,0,1,0,0,0]=>7
[1,0,1,1,1,0,1,1,0,0,0,0]=>8
[1,0,1,1,1,1,0,0,0,0,1,0]=>6
[1,0,1,1,1,1,0,0,0,1,0,0]=>7
[1,0,1,1,1,1,0,0,1,0,0,0]=>8
[1,0,1,1,1,1,0,1,0,0,0,0]=>9
[1,0,1,1,1,1,1,0,0,0,0,0]=>10
[1,1,0,0,1,0,1,0,1,0,1,0]=>1
[1,1,0,0,1,0,1,0,1,1,0,0]=>2
[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]=>4
[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]=>3
[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]=>4
[1,1,0,0,1,1,0,1,1,0,0,0]=>5
[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]=>5
[1,1,0,0,1,1,1,0,1,0,0,0]=>6
[1,1,0,0,1,1,1,1,0,0,0,0]=>7
[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]=>4
[1,1,0,1,0,0,1,1,1,0,0,0]=>5
[1,1,0,1,0,1,0,0,1,0,1,0]=>3
[1,1,0,1,0,1,0,0,1,1,0,0]=>4
[1,1,0,1,0,1,0,1,0,0,1,0]=>4
[1,1,0,1,0,1,0,1,0,1,0,0]=>5
[1,1,0,1,0,1,0,1,1,0,0,0]=>6
[1,1,0,1,0,1,1,0,0,0,1,0]=>5
[1,1,0,1,0,1,1,0,0,1,0,0]=>6
[1,1,0,1,0,1,1,0,1,0,0,0]=>7
[1,1,0,1,0,1,1,1,0,0,0,0]=>8
[1,1,0,1,1,0,0,0,1,0,1,0]=>4
[1,1,0,1,1,0,0,0,1,1,0,0]=>5
[1,1,0,1,1,0,0,1,0,0,1,0]=>5
[1,1,0,1,1,0,0,1,0,1,0,0]=>6
[1,1,0,1,1,0,0,1,1,0,0,0]=>7
[1,1,0,1,1,0,1,0,0,0,1,0]=>6
[1,1,0,1,1,0,1,0,0,1,0,0]=>7
[1,1,0,1,1,0,1,0,1,0,0,0]=>8
[1,1,0,1,1,0,1,1,0,0,0,0]=>9
[1,1,0,1,1,1,0,0,0,0,1,0]=>7
[1,1,0,1,1,1,0,0,0,1,0,0]=>8
[1,1,0,1,1,1,0,0,1,0,0,0]=>9
[1,1,0,1,1,1,0,1,0,0,0,0]=>10
[1,1,0,1,1,1,1,0,0,0,0,0]=>11
[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]=>5
[1,1,1,0,0,0,1,1,1,0,0,0]=>6
[1,1,1,0,0,1,0,0,1,0,1,0]=>4
[1,1,1,0,0,1,0,0,1,1,0,0]=>5
[1,1,1,0,0,1,0,1,0,0,1,0]=>5
[1,1,1,0,0,1,0,1,0,1,0,0]=>6
[1,1,1,0,0,1,0,1,1,0,0,0]=>7
[1,1,1,0,0,1,1,0,0,0,1,0]=>6
[1,1,1,0,0,1,1,0,0,1,0,0]=>7
[1,1,1,0,0,1,1,0,1,0,0,0]=>8
[1,1,1,0,0,1,1,1,0,0,0,0]=>9
[1,1,1,0,1,0,0,0,1,0,1,0]=>5
[1,1,1,0,1,0,0,0,1,1,0,0]=>6
[1,1,1,0,1,0,0,1,0,0,1,0]=>6
[1,1,1,0,1,0,0,1,0,1,0,0]=>7
[1,1,1,0,1,0,0,1,1,0,0,0]=>8
[1,1,1,0,1,0,1,0,0,0,1,0]=>7
[1,1,1,0,1,0,1,0,0,1,0,0]=>8
[1,1,1,0,1,0,1,0,1,0,0,0]=>9
[1,1,1,0,1,0,1,1,0,0,0,0]=>10
[1,1,1,0,1,1,0,0,0,0,1,0]=>8
[1,1,1,0,1,1,0,0,0,1,0,0]=>9
[1,1,1,0,1,1,0,0,1,0,0,0]=>10
[1,1,1,0,1,1,0,1,0,0,0,0]=>11
[1,1,1,0,1,1,1,0,0,0,0,0]=>12
[1,1,1,1,0,0,0,0,1,0,1,0]=>6
[1,1,1,1,0,0,0,0,1,1,0,0]=>7
[1,1,1,1,0,0,0,1,0,0,1,0]=>7
[1,1,1,1,0,0,0,1,0,1,0,0]=>8
[1,1,1,1,0,0,0,1,1,0,0,0]=>9
[1,1,1,1,0,0,1,0,0,0,1,0]=>8
[1,1,1,1,0,0,1,0,0,1,0,0]=>9
[1,1,1,1,0,0,1,0,1,0,0,0]=>10
[1,1,1,1,0,0,1,1,0,0,0,0]=>11
[1,1,1,1,0,1,0,0,0,0,1,0]=>9
[1,1,1,1,0,1,0,0,0,1,0,0]=>10
[1,1,1,1,0,1,0,0,1,0,0,0]=>11
[1,1,1,1,0,1,0,1,0,0,0,0]=>12
[1,1,1,1,0,1,1,0,0,0,0,0]=>13
[1,1,1,1,1,0,0,0,0,0,1,0]=>10
[1,1,1,1,1,0,0,0,0,1,0,0]=>11
[1,1,1,1,1,0,0,0,1,0,0,0]=>12
[1,1,1,1,1,0,0,1,0,0,0,0]=>13
[1,1,1,1,1,0,1,0,0,0,0,0]=>14
[1,1,1,1,1,1,0,0,0,0,0,0]=>15
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Description
The vector space dimension of $\operatorname{Hom}_A(J^2,J^2)$ for the linear Nakayama algebra $A$ corresponding to a Dyck path.
Here $J$ denotes the Jacobson radical of $A$.
The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
Here $J$ denotes the Jacobson radical of $A$.
The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
Code
gap('LoadPackage("QPA");')
import tempfile as _tf, os as _os
_gap_code = r"""
DeclareOperation("hom1rads2", [IsList]);
InstallMethod(hom1rads2, "for a representation of a quiver", [IsList],0,function(L)
local A, J1, J2, RegA;
A := L[1];
RegA := DirectSumOfQPAModules(IndecProjectiveModules(A));
J1 := RadicalOfModule(RegA);
J2 := RadicalOfModule(J1);
return(Size(HomOverAlgebra(J2,J2)));
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)
A = gap.NakayamaAlgebra(gap.GF(3), K)
return ZZ(gap.hom1rads2([A]))
Created
Jul 20, 2018 at 18:31 by Rene Marczinzik
Updated
Mar 13, 2026 at 15:03 by Nupur Jain
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