edit this statistic or download as text // json
Identifier
Values
=>
Cc0005;cc-rep
[1,0]=>0 [1,0,1,0]=>1 [1,1,0,0]=>0 [1,0,1,0,1,0]=>2 [1,0,1,1,0,0]=>1 [1,1,0,0,1,0]=>1 [1,1,0,1,0,0]=>1 [1,1,1,0,0,0]=>0 [1,0,1,0,1,0,1,0]=>3 [1,0,1,0,1,1,0,0]=>2 [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]=>1 [1,1,0,0,1,0,1,0]=>2 [1,1,0,0,1,1,0,0]=>1 [1,1,0,1,0,0,1,0]=>1 [1,1,0,1,0,1,0,0]=>2 [1,1,0,1,1,0,0,0]=>1 [1,1,1,0,0,0,1,0]=>1 [1,1,1,0,0,1,0,0]=>1 [1,1,1,0,1,0,0,0]=>1 [1,1,1,1,0,0,0,0]=>0 [1,0,1,0,1,0,1,0,1,0]=>4 [1,0,1,0,1,0,1,1,0,0]=>3 [1,0,1,0,1,1,0,0,1,0]=>2 [1,0,1,0,1,1,0,1,0,0]=>3 [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]=>1 [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]=>2 [1,0,1,1,1,0,0,0,1,0]=>1 [1,0,1,1,1,0,0,1,0,0]=>1 [1,0,1,1,1,0,1,0,0,0]=>2 [1,0,1,1,1,1,0,0,0,0]=>1 [1,1,0,0,1,0,1,0,1,0]=>3 [1,1,0,0,1,0,1,1,0,0]=>2 [1,1,0,0,1,1,0,0,1,0]=>1 [1,1,0,0,1,1,0,1,0,0]=>2 [1,1,0,0,1,1,1,0,0,0]=>1 [1,1,0,1,0,0,1,0,1,0]=>2 [1,1,0,1,0,0,1,1,0,0]=>1 [1,1,0,1,0,1,0,0,1,0]=>3 [1,1,0,1,0,1,0,1,0,0]=>2 [1,1,0,1,0,1,1,0,0,0]=>2 [1,1,0,1,1,0,0,0,1,0]=>1 [1,1,0,1,1,0,0,1,0,0]=>1 [1,1,0,1,1,0,1,0,0,0]=>2 [1,1,0,1,1,1,0,0,0,0]=>1 [1,1,1,0,0,0,1,0,1,0]=>2 [1,1,1,0,0,0,1,1,0,0]=>1 [1,1,1,0,0,1,0,0,1,0]=>1 [1,1,1,0,0,1,0,1,0,0]=>2 [1,1,1,0,0,1,1,0,0,0]=>1 [1,1,1,0,1,0,0,0,1,0]=>1 [1,1,1,0,1,0,0,1,0,0]=>1 [1,1,1,0,1,0,1,0,0,0]=>2 [1,1,1,0,1,1,0,0,0,0]=>1 [1,1,1,1,0,0,0,0,1,0]=>1 [1,1,1,1,0,0,0,1,0,0]=>1 [1,1,1,1,0,0,1,0,0,0]=>1 [1,1,1,1,0,1,0,0,0,0]=>1 [1,1,1,1,1,0,0,0,0,0]=>0 [1,0,1,0,1,0,1,0,1,0,1,0]=>5 [1,0,1,0,1,0,1,0,1,1,0,0]=>4 [1,0,1,0,1,0,1,1,0,0,1,0]=>3 [1,0,1,0,1,0,1,1,0,1,0,0]=>4 [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]=>2 [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]=>3 [1,0,1,0,1,1,0,1,0,1,0,0]=>4 [1,0,1,0,1,1,0,1,1,0,0,0]=>3 [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]=>3 [1,0,1,0,1,1,1,1,0,0,0,0]=>2 [1,0,1,1,0,0,1,0,1,0,1,0]=>3 [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]=>1 [1,0,1,1,0,0,1,1,0,1,0,0]=>2 [1,0,1,1,0,0,1,1,1,0,0,0]=>1 [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]=>2 [1,0,1,1,0,1,0,1,0,0,1,0]=>4 [1,0,1,1,0,1,0,1,0,1,0,0]=>3 [1,0,1,1,0,1,0,1,1,0,0,0]=>3 [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]=>3 [1,0,1,1,0,1,1,1,0,0,0,0]=>2 [1,0,1,1,1,0,0,0,1,0,1,0]=>2 [1,0,1,1,1,0,0,0,1,1,0,0]=>1 [1,0,1,1,1,0,0,1,0,0,1,0]=>1 [1,0,1,1,1,0,0,1,0,1,0,0]=>2 [1,0,1,1,1,0,0,1,1,0,0,0]=>1 [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]=>3 [1,0,1,1,1,0,1,1,0,0,0,0]=>2 [1,0,1,1,1,1,0,0,0,0,1,0]=>1 [1,0,1,1,1,1,0,0,0,1,0,0]=>1 [1,0,1,1,1,1,0,0,1,0,0,0]=>1 [1,0,1,1,1,1,0,1,0,0,0,0]=>2 [1,0,1,1,1,1,1,0,0,0,0,0]=>1 [1,1,0,0,1,0,1,0,1,0,1,0]=>4 [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]=>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]=>2 [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]=>1 [1,1,0,0,1,1,0,1,0,0,1,0]=>2 [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]=>2 [1,1,0,0,1,1,1,0,0,0,1,0]=>1 [1,1,0,0,1,1,1,0,0,1,0,0]=>1 [1,1,0,0,1,1,1,0,1,0,0,0]=>2 [1,1,0,0,1,1,1,1,0,0,0,0]=>1 [1,1,0,1,0,0,1,0,1,0,1,0]=>3 [1,1,0,1,0,0,1,0,1,1,0,0]=>2 [1,1,0,1,0,0,1,1,0,0,1,0]=>1 [1,1,0,1,0,0,1,1,0,1,0,0]=>2 [1,1,0,1,0,0,1,1,1,0,0,0]=>1 [1,1,0,1,0,1,0,0,1,0,1,0]=>4 [1,1,0,1,0,1,0,0,1,1,0,0]=>3 [1,1,0,1,0,1,0,1,0,0,1,0]=>3 [1,1,0,1,0,1,0,1,0,1,0,0]=>3 [1,1,0,1,0,1,0,1,1,0,0,0]=>2 [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]=>3 [1,1,0,1,0,1,1,0,1,0,0,0]=>2 [1,1,0,1,0,1,1,1,0,0,0,0]=>2 [1,1,0,1,1,0,0,0,1,0,1,0]=>2 [1,1,0,1,1,0,0,0,1,1,0,0]=>1 [1,1,0,1,1,0,0,1,0,0,1,0]=>1 [1,1,0,1,1,0,0,1,0,1,0,0]=>2 [1,1,0,1,1,0,0,1,1,0,0,0]=>1 [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]=>3 [1,1,0,1,1,0,1,0,1,0,0,0]=>2 [1,1,0,1,1,0,1,1,0,0,0,0]=>2 [1,1,0,1,1,1,0,0,0,0,1,0]=>1 [1,1,0,1,1,1,0,0,0,1,0,0]=>1 [1,1,0,1,1,1,0,0,1,0,0,0]=>1 [1,1,0,1,1,1,0,1,0,0,0,0]=>2 [1,1,0,1,1,1,1,0,0,0,0,0]=>1 [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]=>2 [1,1,1,0,0,0,1,1,0,0,1,0]=>1 [1,1,1,0,0,0,1,1,0,1,0,0]=>2 [1,1,1,0,0,0,1,1,1,0,0,0]=>1 [1,1,1,0,0,1,0,0,1,0,1,0]=>2 [1,1,1,0,0,1,0,0,1,1,0,0]=>1 [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]=>2 [1,1,1,0,0,1,0,1,1,0,0,0]=>2 [1,1,1,0,0,1,1,0,0,0,1,0]=>1 [1,1,1,0,0,1,1,0,0,1,0,0]=>1 [1,1,1,0,0,1,1,0,1,0,0,0]=>2 [1,1,1,0,0,1,1,1,0,0,0,0]=>1 [1,1,1,0,1,0,0,0,1,0,1,0]=>2 [1,1,1,0,1,0,0,0,1,1,0,0]=>1 [1,1,1,0,1,0,0,1,0,0,1,0]=>1 [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]=>1 [1,1,1,0,1,0,1,0,0,0,1,0]=>3 [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]=>2 [1,1,1,0,1,0,1,1,0,0,0,0]=>2 [1,1,1,0,1,1,0,0,0,0,1,0]=>1 [1,1,1,0,1,1,0,0,0,1,0,0]=>1 [1,1,1,0,1,1,0,0,1,0,0,0]=>1 [1,1,1,0,1,1,0,1,0,0,0,0]=>2 [1,1,1,0,1,1,1,0,0,0,0,0]=>1 [1,1,1,1,0,0,0,0,1,0,1,0]=>2 [1,1,1,1,0,0,0,0,1,1,0,0]=>1 [1,1,1,1,0,0,0,1,0,0,1,0]=>1 [1,1,1,1,0,0,0,1,0,1,0,0]=>2 [1,1,1,1,0,0,0,1,1,0,0,0]=>1 [1,1,1,1,0,0,1,0,0,0,1,0]=>1 [1,1,1,1,0,0,1,0,0,1,0,0]=>1 [1,1,1,1,0,0,1,0,1,0,0,0]=>2 [1,1,1,1,0,0,1,1,0,0,0,0]=>1 [1,1,1,1,0,1,0,0,0,0,1,0]=>1 [1,1,1,1,0,1,0,0,0,1,0,0]=>1 [1,1,1,1,0,1,0,0,1,0,0,0]=>1 [1,1,1,1,0,1,0,1,0,0,0,0]=>2 [1,1,1,1,0,1,1,0,0,0,0,0]=>1 [1,1,1,1,1,0,0,0,0,0,1,0]=>1 [1,1,1,1,1,0,0,0,0,1,0,0]=>1 [1,1,1,1,1,0,0,0,1,0,0,0]=>1 [1,1,1,1,1,0,0,1,0,0,0,0]=>1 [1,1,1,1,1,0,1,0,0,0,0,0]=>1 [1,1,1,1,1,1,0,0,0,0,0,0]=>0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The maximal torsionfree index of a simple non-projective module of the linear Nakayama algebra corresponding to a Dyck path.
The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
The number of algebras where the statistic returns a value less than or equal to one might be given by the Motzkin numbers oeis.org/A001006.
Code
gap('LoadPackage("QPA");')

import tempfile as _tf, os as _os
_gap_code = r"""
DeclareOperation("IsNtorsionfree",[IsList]);

InstallMethod(IsNtorsionfree, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, CoRegA, M, i, n, temm23;
    A := LIST[1];
    M := LIST[2];

    n := LIST[3];

    CoRegA := DirectSumOfQPAModules(IndecInjectiveModules(A));
    temm23 := [];
    for i in [0..n-1] do Append(temm23,[Size(ExtOverAlgebra(NthSyzygy(CoRegA,i),DTr(M))[2])]);
    od;
    return(Sum(temm23));
end);

DeclareOperation("torsionfreeindex",[IsList]);

InstallMethod(torsionfreeindex, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, M, U, g;
    A := LIST[1];
    M := LIST[2];
    g := LIST[3];
    U := Filtered([1..g],x->IsNtorsionfree([A,M,x])>0);
    return(Minimum(U)-1);
end);

DeclareOperation("torsionmax",[IsList]);

InstallMethod(torsionmax, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, UU, g, i, simA;
    A := LIST[1];
    g := GlobalDimensionOfAlgebra(A,30);
    simA := Filtered(SimpleModules(A),x->IsProjectiveModule(x)=false);
    UU := [];
    for i in simA do Append(UU,[torsionfreeindex([A,i,g])]);
    od;
    return(Maximum(UU));
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.torsionmax([A]))
Created
Nov 24, 2018 at 16:21 by Rene Marczinzik
Updated
Mar 13, 2026 at 15:04 by Nupur Jain