Identifier
- St001138: Dyck paths ⟶ ℤ
Values
=>
Cc0005;cc-rep
[1,0]=>3
[1,0,1,0]=>5
[1,1,0,0]=>6
[1,0,1,0,1,0]=>7
[1,0,1,1,0,0]=>8
[1,1,0,0,1,0]=>8
[1,1,0,1,0,0]=>9
[1,1,1,0,0,0]=>10
[1,0,1,0,1,0,1,0]=>8
[1,0,1,0,1,1,0,0]=>10
[1,0,1,1,0,0,1,0]=>9
[1,0,1,1,0,1,0,0]=>11
[1,0,1,1,1,0,0,0]=>12
[1,1,0,0,1,0,1,0]=>10
[1,1,0,0,1,1,0,0]=>11
[1,1,0,1,0,0,1,0]=>11
[1,1,0,1,0,1,0,0]=>12
[1,1,0,1,1,0,0,0]=>13
[1,1,1,0,0,0,1,0]=>12
[1,1,1,0,0,1,0,0]=>13
[1,1,1,0,1,0,0,0]=>14
[1,1,1,1,0,0,0,0]=>15
[1,0,1,0,1,0,1,0,1,0]=>9
[1,0,1,0,1,0,1,1,0,0]=>11
[1,0,1,0,1,1,0,0,1,0]=>11
[1,0,1,0,1,1,0,1,0,0]=>12
[1,0,1,0,1,1,1,0,0,0]=>14
[1,0,1,1,0,0,1,0,1,0]=>11
[1,0,1,1,0,0,1,1,0,0]=>12
[1,0,1,1,0,1,0,0,1,0]=>12
[1,0,1,1,0,1,0,1,0,0]=>12
[1,0,1,1,0,1,1,0,0,0]=>15
[1,0,1,1,1,0,0,0,1,0]=>13
[1,0,1,1,1,0,0,1,0,0]=>13
[1,0,1,1,1,0,1,0,0,0]=>16
[1,0,1,1,1,1,0,0,0,0]=>17
[1,1,0,0,1,0,1,0,1,0]=>11
[1,1,0,0,1,0,1,1,0,0]=>13
[1,1,0,0,1,1,0,0,1,0]=>12
[1,1,0,0,1,1,0,1,0,0]=>14
[1,1,0,0,1,1,1,0,0,0]=>15
[1,1,0,1,0,0,1,0,1,0]=>12
[1,1,0,1,0,0,1,1,0,0]=>14
[1,1,0,1,0,1,0,0,1,0]=>12
[1,1,0,1,0,1,0,1,0,0]=>14
[1,1,0,1,0,1,1,0,0,0]=>16
[1,1,0,1,1,0,0,0,1,0]=>13
[1,1,0,1,1,0,0,1,0,0]=>15
[1,1,0,1,1,0,1,0,0,0]=>17
[1,1,0,1,1,1,0,0,0,0]=>18
[1,1,1,0,0,0,1,0,1,0]=>14
[1,1,1,0,0,0,1,1,0,0]=>15
[1,1,1,0,0,1,0,0,1,0]=>15
[1,1,1,0,0,1,0,1,0,0]=>16
[1,1,1,0,0,1,1,0,0,0]=>17
[1,1,1,0,1,0,0,0,1,0]=>16
[1,1,1,0,1,0,0,1,0,0]=>17
[1,1,1,0,1,0,1,0,0,0]=>18
[1,1,1,0,1,1,0,0,0,0]=>19
[1,1,1,1,0,0,0,0,1,0]=>17
[1,1,1,1,0,0,0,1,0,0]=>18
[1,1,1,1,0,0,1,0,0,0]=>19
[1,1,1,1,0,1,0,0,0,0]=>20
[1,1,1,1,1,0,0,0,0,0]=>21
[1,0,1,0,1,0,1,0,1,0,1,0]=>10
[1,0,1,0,1,0,1,0,1,1,0,0]=>12
[1,0,1,0,1,0,1,1,0,0,1,0]=>12
[1,0,1,0,1,0,1,1,0,1,0,0]=>13
[1,0,1,0,1,0,1,1,1,0,0,0]=>15
[1,0,1,0,1,1,0,0,1,0,1,0]=>13
[1,0,1,0,1,1,0,0,1,1,0,0]=>14
[1,0,1,0,1,1,0,1,0,0,1,0]=>13
[1,0,1,0,1,1,0,1,0,1,0,0]=>13
[1,0,1,0,1,1,0,1,1,0,0,0]=>16
[1,0,1,0,1,1,1,0,0,0,1,0]=>15
[1,0,1,0,1,1,1,0,0,1,0,0]=>15
[1,0,1,0,1,1,1,0,1,0,0,0]=>17
[1,0,1,0,1,1,1,1,0,0,0,0]=>19
[1,0,1,1,0,0,1,0,1,0,1,0]=>12
[1,0,1,1,0,0,1,0,1,1,0,0]=>14
[1,0,1,1,0,0,1,1,0,0,1,0]=>13
[1,0,1,1,0,0,1,1,0,1,0,0]=>15
[1,0,1,1,0,0,1,1,1,0,0,0]=>16
[1,0,1,1,0,1,0,0,1,0,1,0]=>13
[1,0,1,1,0,1,0,0,1,1,0,0]=>15
[1,0,1,1,0,1,0,1,0,0,1,0]=>12
[1,0,1,1,0,1,0,1,0,1,0,0]=>14
[1,0,1,1,0,1,0,1,1,0,0,0]=>16
[1,0,1,1,0,1,1,0,0,0,1,0]=>15
[1,0,1,1,0,1,1,0,0,1,0,0]=>16
[1,0,1,1,0,1,1,0,1,0,0,0]=>17
[1,0,1,1,0,1,1,1,0,0,0,0]=>20
[1,0,1,1,1,0,0,0,1,0,1,0]=>15
[1,0,1,1,1,0,0,0,1,1,0,0]=>16
[1,0,1,1,1,0,0,1,0,0,1,0]=>15
[1,0,1,1,1,0,0,1,0,1,0,0]=>16
[1,0,1,1,1,0,0,1,1,0,0,0]=>17
[1,0,1,1,1,0,1,0,0,0,1,0]=>17
[1,0,1,1,1,0,1,0,0,1,0,0]=>17
[1,0,1,1,1,0,1,0,1,0,0,0]=>17
[1,0,1,1,1,0,1,1,0,0,0,0]=>21
[1,0,1,1,1,1,0,0,0,0,1,0]=>18
[1,0,1,1,1,1,0,0,0,1,0,0]=>18
[1,0,1,1,1,1,0,0,1,0,0,0]=>18
[1,0,1,1,1,1,0,1,0,0,0,0]=>22
[1,0,1,1,1,1,1,0,0,0,0,0]=>23
[1,1,0,0,1,0,1,0,1,0,1,0]=>12
[1,1,0,0,1,0,1,0,1,1,0,0]=>14
[1,1,0,0,1,0,1,1,0,0,1,0]=>14
[1,1,0,0,1,0,1,1,0,1,0,0]=>15
[1,1,0,0,1,0,1,1,1,0,0,0]=>17
[1,1,0,0,1,1,0,0,1,0,1,0]=>14
[1,1,0,0,1,1,0,0,1,1,0,0]=>15
[1,1,0,0,1,1,0,1,0,0,1,0]=>15
[1,1,0,0,1,1,0,1,0,1,0,0]=>15
[1,1,0,0,1,1,0,1,1,0,0,0]=>18
[1,1,0,0,1,1,1,0,0,0,1,0]=>16
[1,1,0,0,1,1,1,0,0,1,0,0]=>16
[1,1,0,0,1,1,1,0,1,0,0,0]=>19
[1,1,0,0,1,1,1,1,0,0,0,0]=>20
[1,1,0,1,0,0,1,0,1,0,1,0]=>13
[1,1,0,1,0,0,1,0,1,1,0,0]=>15
[1,1,0,1,0,0,1,1,0,0,1,0]=>15
[1,1,0,1,0,0,1,1,0,1,0,0]=>16
[1,1,0,1,0,0,1,1,1,0,0,0]=>18
[1,1,0,1,0,1,0,0,1,0,1,0]=>13
[1,1,0,1,0,1,0,0,1,1,0,0]=>15
[1,1,0,1,0,1,0,1,0,0,1,0]=>14
[1,1,0,1,0,1,0,1,0,1,0,0]=>15
[1,1,0,1,0,1,0,1,1,0,0,0]=>18
[1,1,0,1,0,1,1,0,0,0,1,0]=>16
[1,1,0,1,0,1,1,0,0,1,0,0]=>16
[1,1,0,1,0,1,1,0,1,0,0,0]=>19
[1,1,0,1,0,1,1,1,0,0,0,0]=>21
[1,1,0,1,1,0,0,0,1,0,1,0]=>15
[1,1,0,1,1,0,0,0,1,1,0,0]=>16
[1,1,0,1,1,0,0,1,0,0,1,0]=>16
[1,1,0,1,1,0,0,1,0,1,0,0]=>16
[1,1,0,1,1,0,0,1,1,0,0,0]=>19
[1,1,0,1,1,0,1,0,0,0,1,0]=>17
[1,1,0,1,1,0,1,0,0,1,0,0]=>16
[1,1,0,1,1,0,1,0,1,0,0,0]=>19
[1,1,0,1,1,0,1,1,0,0,0,0]=>22
[1,1,0,1,1,1,0,0,0,0,1,0]=>18
[1,1,0,1,1,1,0,0,0,1,0,0]=>17
[1,1,0,1,1,1,0,0,1,0,0,0]=>20
[1,1,0,1,1,1,0,1,0,0,0,0]=>23
[1,1,0,1,1,1,1,0,0,0,0,0]=>24
[1,1,1,0,0,0,1,0,1,0,1,0]=>15
[1,1,1,0,0,0,1,0,1,1,0,0]=>17
[1,1,1,0,0,0,1,1,0,0,1,0]=>16
[1,1,1,0,0,0,1,1,0,1,0,0]=>18
[1,1,1,0,0,0,1,1,1,0,0,0]=>19
[1,1,1,0,0,1,0,0,1,0,1,0]=>16
[1,1,1,0,0,1,0,0,1,1,0,0]=>18
[1,1,1,0,0,1,0,1,0,0,1,0]=>16
[1,1,1,0,0,1,0,1,0,1,0,0]=>18
[1,1,1,0,0,1,0,1,1,0,0,0]=>20
[1,1,1,0,0,1,1,0,0,0,1,0]=>17
[1,1,1,0,0,1,1,0,0,1,0,0]=>19
[1,1,1,0,0,1,1,0,1,0,0,0]=>21
[1,1,1,0,0,1,1,1,0,0,0,0]=>22
[1,1,1,0,1,0,0,0,1,0,1,0]=>17
[1,1,1,0,1,0,0,0,1,1,0,0]=>19
[1,1,1,0,1,0,0,1,0,0,1,0]=>17
[1,1,1,0,1,0,0,1,0,1,0,0]=>19
[1,1,1,0,1,0,0,1,1,0,0,0]=>21
[1,1,1,0,1,0,1,0,0,0,1,0]=>17
[1,1,1,0,1,0,1,0,0,1,0,0]=>19
[1,1,1,0,1,0,1,0,1,0,0,0]=>21
[1,1,1,0,1,0,1,1,0,0,0,0]=>23
[1,1,1,0,1,1,0,0,0,0,1,0]=>18
[1,1,1,0,1,1,0,0,0,1,0,0]=>20
[1,1,1,0,1,1,0,0,1,0,0,0]=>22
[1,1,1,0,1,1,0,1,0,0,0,0]=>24
[1,1,1,0,1,1,1,0,0,0,0,0]=>25
[1,1,1,1,0,0,0,0,1,0,1,0]=>19
[1,1,1,1,0,0,0,0,1,1,0,0]=>20
[1,1,1,1,0,0,0,1,0,0,1,0]=>20
[1,1,1,1,0,0,0,1,0,1,0,0]=>21
[1,1,1,1,0,0,0,1,1,0,0,0]=>22
[1,1,1,1,0,0,1,0,0,0,1,0]=>21
[1,1,1,1,0,0,1,0,0,1,0,0]=>22
[1,1,1,1,0,0,1,0,1,0,0,0]=>23
[1,1,1,1,0,0,1,1,0,0,0,0]=>24
[1,1,1,1,0,1,0,0,0,0,1,0]=>22
[1,1,1,1,0,1,0,0,0,1,0,0]=>23
[1,1,1,1,0,1,0,0,1,0,0,0]=>24
[1,1,1,1,0,1,0,1,0,0,0,0]=>25
[1,1,1,1,0,1,1,0,0,0,0,0]=>26
[1,1,1,1,1,0,0,0,0,0,1,0]=>23
[1,1,1,1,1,0,0,0,0,1,0,0]=>24
[1,1,1,1,1,0,0,0,1,0,0,0]=>25
[1,1,1,1,1,0,0,1,0,0,0,0]=>26
[1,1,1,1,1,0,1,0,0,0,0,0]=>27
[1,1,1,1,1,1,0,0,0,0,0,0]=>28
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Description
The number of indecomposable modules of the linear Nakayama algebra corresponding to a Dyck path whose projective dimension or injective dimension is at most one.
The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
References
[1] Marczinzik, René Upper bounds for the dominant dimension of Nakayama and related algebras. zbMATH:06820683
Code
gap('LoadPackage("QPA");')
import tempfile as _tf, os as _os
_gap_code = r"""
DeclareOperation("ARQuiver", [IsList]);
InstallMethod(ARQuiver, "for an algebra and bound", [IsList], 0, function(LIST)
local A, L, N, PI, bound, dim, f, h, inj, injA, j;
A := LIST[1];
bound := LIST[2];
injA := IndecInjectiveModules(A);
L := [];
for inj in injA do
dim := Dimension(inj);
for j in [0..dim-1] do
if j = 0 then
f := IdentityMapping(inj);
else
f := RadicalOfModuleInclusion(inj);
N := Source(f);
h := 1;
while h < j do
f := RadicalOfModuleInclusion(N);
N := Source(f);
h := h + 1;
od;
fi;
Add(L, Source(f));
od;
od;
PI := Filtered(L, x -> IsProjectiveModule(x) and IsInjectiveModule(x));
return [PI, L];
end);
DeclareOperation("numberofmoduleswithprojinjdimlessorequal1",[IsList]);
InstallMethod(numberofmoduleswithprojinjdimlessorequal1, "for a representation of a quiver", [IsList],0,function(LIST)
local A, L, LL, LU;
LU := LIST[1];
A := NakayamaAlgebra(LU,GF(3));
L := ARQuiver([A,1000])[2];
LL := Filtered(L,x->ProjDimensionOfModule(x,30)<=1 or InjDimensionOfModule(x,30)<=1);
return(Size(LL));
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.numberofmoduleswithprojinjdimlessorequal1([K]))
Created
Apr 09, 2018 at 14:03 by Rene Marczinzik
Updated
Mar 12, 2026 at 16:00 by Nupur Jain
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