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Identifier
Values
=>
Cc0005;cc-rep
[1,0]=>2 [1,0,1,0]=>3 [1,1,0,0]=>3 [1,0,1,0,1,0]=>4 [1,0,1,1,0,0]=>4 [1,1,0,0,1,0]=>4 [1,1,0,1,0,0]=>4 [1,1,1,0,0,0]=>4 [1,0,1,0,1,0,1,0]=>4 [1,0,1,0,1,1,0,0]=>5 [1,0,1,1,0,0,1,0]=>5 [1,0,1,1,0,1,0,0]=>5 [1,0,1,1,1,0,0,0]=>5 [1,1,0,0,1,0,1,0]=>5 [1,1,0,0,1,1,0,0]=>5 [1,1,0,1,0,0,1,0]=>5 [1,1,0,1,0,1,0,0]=>5 [1,1,0,1,1,0,0,0]=>5 [1,1,1,0,0,0,1,0]=>5 [1,1,1,0,0,1,0,0]=>5 [1,1,1,0,1,0,0,0]=>5 [1,1,1,1,0,0,0,0]=>5 [1,0,1,0,1,0,1,0,1,0]=>4 [1,0,1,0,1,0,1,1,0,0]=>5 [1,0,1,0,1,1,0,0,1,0]=>6 [1,0,1,0,1,1,0,1,0,0]=>5 [1,0,1,0,1,1,1,0,0,0]=>6 [1,0,1,1,0,0,1,0,1,0]=>6 [1,0,1,1,0,0,1,1,0,0]=>6 [1,0,1,1,0,1,0,0,1,0]=>5 [1,0,1,1,0,1,0,1,0,0]=>5 [1,0,1,1,0,1,1,0,0,0]=>6 [1,0,1,1,1,0,0,0,1,0]=>6 [1,0,1,1,1,0,0,1,0,0]=>6 [1,0,1,1,1,0,1,0,0,0]=>6 [1,0,1,1,1,1,0,0,0,0]=>6 [1,1,0,0,1,0,1,0,1,0]=>5 [1,1,0,0,1,0,1,1,0,0]=>6 [1,1,0,0,1,1,0,0,1,0]=>6 [1,1,0,0,1,1,0,1,0,0]=>6 [1,1,0,0,1,1,1,0,0,0]=>6 [1,1,0,1,0,0,1,0,1,0]=>5 [1,1,0,1,0,0,1,1,0,0]=>6 [1,1,0,1,0,1,0,0,1,0]=>5 [1,1,0,1,0,1,0,1,0,0]=>6 [1,1,0,1,0,1,1,0,0,0]=>6 [1,1,0,1,1,0,0,0,1,0]=>6 [1,1,0,1,1,0,0,1,0,0]=>6 [1,1,0,1,1,0,1,0,0,0]=>6 [1,1,0,1,1,1,0,0,0,0]=>6 [1,1,1,0,0,0,1,0,1,0]=>6 [1,1,1,0,0,0,1,1,0,0]=>6 [1,1,1,0,0,1,0,0,1,0]=>6 [1,1,1,0,0,1,0,1,0,0]=>6 [1,1,1,0,0,1,1,0,0,0]=>6 [1,1,1,0,1,0,0,0,1,0]=>6 [1,1,1,0,1,0,0,1,0,0]=>6 [1,1,1,0,1,0,1,0,0,0]=>6 [1,1,1,0,1,1,0,0,0,0]=>6 [1,1,1,1,0,0,0,0,1,0]=>6 [1,1,1,1,0,0,0,1,0,0]=>6 [1,1,1,1,0,0,1,0,0,0]=>6 [1,1,1,1,0,1,0,0,0,0]=>6 [1,1,1,1,1,0,0,0,0,0]=>6 [1,0,1,0,1,0,1,0,1,0,1,0]=>4 [1,0,1,0,1,0,1,0,1,1,0,0]=>5 [1,0,1,0,1,0,1,1,0,0,1,0]=>6 [1,0,1,0,1,0,1,1,0,1,0,0]=>5 [1,0,1,0,1,0,1,1,1,0,0,0]=>6 [1,0,1,0,1,1,0,0,1,0,1,0]=>7 [1,0,1,0,1,1,0,0,1,1,0,0]=>7 [1,0,1,0,1,1,0,1,0,0,1,0]=>5 [1,0,1,0,1,1,0,1,0,1,0,0]=>5 [1,0,1,0,1,1,0,1,1,0,0,0]=>6 [1,0,1,0,1,1,1,0,0,0,1,0]=>7 [1,0,1,0,1,1,1,0,0,1,0,0]=>7 [1,0,1,0,1,1,1,0,1,0,0,0]=>6 [1,0,1,0,1,1,1,1,0,0,0,0]=>7 [1,0,1,1,0,0,1,0,1,0,1,0]=>6 [1,0,1,1,0,0,1,0,1,1,0,0]=>7 [1,0,1,1,0,0,1,1,0,0,1,0]=>7 [1,0,1,1,0,0,1,1,0,1,0,0]=>7 [1,0,1,1,0,0,1,1,1,0,0,0]=>7 [1,0,1,1,0,1,0,0,1,0,1,0]=>5 [1,0,1,1,0,1,0,0,1,1,0,0]=>6 [1,0,1,1,0,1,0,1,0,0,1,0]=>5 [1,0,1,1,0,1,0,1,0,1,0,0]=>6 [1,0,1,1,0,1,0,1,1,0,0,0]=>6 [1,0,1,1,0,1,1,0,0,0,1,0]=>7 [1,0,1,1,0,1,1,0,0,1,0,0]=>6 [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]=>7 [1,0,1,1,1,0,0,0,1,1,0,0]=>7 [1,0,1,1,1,0,0,1,0,0,1,0]=>7 [1,0,1,1,1,0,0,1,0,1,0,0]=>7 [1,0,1,1,1,0,0,1,1,0,0,0]=>7 [1,0,1,1,1,0,1,0,0,0,1,0]=>6 [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]=>6 [1,0,1,1,1,0,1,1,0,0,0,0]=>7 [1,0,1,1,1,1,0,0,0,0,1,0]=>7 [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]=>7 [1,0,1,1,1,1,0,1,0,0,0,0]=>7 [1,0,1,1,1,1,1,0,0,0,0,0]=>7 [1,1,0,0,1,0,1,0,1,0,1,0]=>5 [1,1,0,0,1,0,1,0,1,1,0,0]=>6 [1,1,0,0,1,0,1,1,0,0,1,0]=>7 [1,1,0,0,1,0,1,1,0,1,0,0]=>6 [1,1,0,0,1,0,1,1,1,0,0,0]=>7 [1,1,0,0,1,1,0,0,1,0,1,0]=>7 [1,1,0,0,1,1,0,0,1,1,0,0]=>7 [1,1,0,0,1,1,0,1,0,0,1,0]=>6 [1,1,0,0,1,1,0,1,0,1,0,0]=>6 [1,1,0,0,1,1,0,1,1,0,0,0]=>7 [1,1,0,0,1,1,1,0,0,0,1,0]=>7 [1,1,0,0,1,1,1,0,0,1,0,0]=>7 [1,1,0,0,1,1,1,0,1,0,0,0]=>7 [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]=>5 [1,1,0,1,0,0,1,0,1,1,0,0]=>6 [1,1,0,1,0,0,1,1,0,0,1,0]=>7 [1,1,0,1,0,0,1,1,0,1,0,0]=>6 [1,1,0,1,0,0,1,1,1,0,0,0]=>7 [1,1,0,1,0,1,0,0,1,0,1,0]=>5 [1,1,0,1,0,1,0,0,1,1,0,0]=>6 [1,1,0,1,0,1,0,1,0,0,1,0]=>6 [1,1,0,1,0,1,0,1,0,1,0,0]=>6 [1,1,0,1,0,1,0,1,1,0,0,0]=>7 [1,1,0,1,0,1,1,0,0,0,1,0]=>7 [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]=>7 [1,1,0,1,1,0,0,0,1,0,1,0]=>7 [1,1,0,1,1,0,0,0,1,1,0,0]=>7 [1,1,0,1,1,0,0,1,0,0,1,0]=>6 [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]=>6 [1,1,0,1,1,0,1,0,1,0,0,0]=>7 [1,1,0,1,1,0,1,1,0,0,0,0]=>7 [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]=>7 [1,1,0,1,1,1,0,0,1,0,0,0]=>7 [1,1,0,1,1,1,0,1,0,0,0,0]=>7 [1,1,0,1,1,1,1,0,0,0,0,0]=>7 [1,1,1,0,0,0,1,0,1,0,1,0]=>6 [1,1,1,0,0,0,1,0,1,1,0,0]=>7 [1,1,1,0,0,0,1,1,0,0,1,0]=>7 [1,1,1,0,0,0,1,1,0,1,0,0]=>7 [1,1,1,0,0,0,1,1,1,0,0,0]=>7 [1,1,1,0,0,1,0,0,1,0,1,0]=>6 [1,1,1,0,0,1,0,0,1,1,0,0]=>7 [1,1,1,0,0,1,0,1,0,0,1,0]=>6 [1,1,1,0,0,1,0,1,0,1,0,0]=>7 [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]=>7 [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]=>7 [1,1,1,0,0,1,1,1,0,0,0,0]=>7 [1,1,1,0,1,0,0,0,1,0,1,0]=>6 [1,1,1,0,1,0,0,0,1,1,0,0]=>7 [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]=>7 [1,1,1,0,1,0,1,0,0,0,1,0]=>6 [1,1,1,0,1,0,1,0,0,1,0,0]=>7 [1,1,1,0,1,0,1,0,1,0,0,0]=>7 [1,1,1,0,1,0,1,1,0,0,0,0]=>7 [1,1,1,0,1,1,0,0,0,0,1,0]=>7 [1,1,1,0,1,1,0,0,0,1,0,0]=>7 [1,1,1,0,1,1,0,0,1,0,0,0]=>7 [1,1,1,0,1,1,0,1,0,0,0,0]=>7 [1,1,1,0,1,1,1,0,0,0,0,0]=>7 [1,1,1,1,0,0,0,0,1,0,1,0]=>7 [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]=>7 [1,1,1,1,0,0,0,1,1,0,0,0]=>7 [1,1,1,1,0,0,1,0,0,0,1,0]=>7 [1,1,1,1,0,0,1,0,0,1,0,0]=>7 [1,1,1,1,0,0,1,0,1,0,0,0]=>7 [1,1,1,1,0,0,1,1,0,0,0,0]=>7 [1,1,1,1,0,1,0,0,0,0,1,0]=>7 [1,1,1,1,0,1,0,0,0,1,0,0]=>7 [1,1,1,1,0,1,0,0,1,0,0,0]=>7 [1,1,1,1,0,1,0,1,0,0,0,0]=>7 [1,1,1,1,0,1,1,0,0,0,0,0]=>7 [1,1,1,1,1,0,0,0,0,0,1,0]=>7 [1,1,1,1,1,0,0,0,0,1,0,0]=>7 [1,1,1,1,1,0,0,0,1,0,0,0]=>7 [1,1,1,1,1,0,0,1,0,0,0,0]=>7 [1,1,1,1,1,0,1,0,0,0,0,0]=>7 [1,1,1,1,1,1,0,0,0,0,0,0]=>7
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Description
The number of simple modules with projective dimension at most three in the linear Nakayama algebra corresponding to a Dyck path.
See St001007The number of simple modules with projective dimension one in the linear Nakayama algebra corresponding to a Dyck path., St001011The number of simple modules with projective dimension two in the linear Nakayama algebra corresponding to a Dyck path., and St001022The number of simple modules with projective dimension three in the linear Nakayama algebra corresponding to a Dyck path. for the number of simple modules with projective dimension one, two, and three respectively.
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("numberssimpprojdimatmost3", [IsList]);

InstallMethod(numberssimpprojdimatmost3, "for a representation of a quiver", [IsList],0,function(L)
    local A, R, RR, list;
    list := L;

    A := NakayamaAlgebra(GF(3),list);
    R := SimpleModules(A);
    RR := Filtered(R,x->ProjDimensionOfModule(x,3)<=3);
    return(Size(RR));
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.numberssimpprojdimatmost3(K))
Created
Oct 30, 2017 at 21:48 by Rene Marczinzik
Updated
Mar 12, 2026 at 14:52 by Nupur Jain