view pending changes or download as text // json
Identifier
Values
=>
Cc0005;cc-rep
[1,0]=>1 [1,0,1,0]=>2 [1,1,0,0]=>1 [1,0,1,0,1,0]=>3 [1,0,1,1,0,0]=>2 [1,1,0,0,1,0]=>2 [1,1,0,1,0,0]=>1 [1,1,1,0,0,0]=>1 [1,0,1,0,1,0,1,0]=>4 [1,0,1,0,1,1,0,0]=>3 [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]=>2 [1,1,0,0,1,0,1,0]=>3 [1,1,0,0,1,1,0,0]=>2 [1,1,0,1,0,0,1,0]=>1 [1,1,0,1,0,1,0,0]=>3 [1,1,0,1,1,0,0,0]=>1 [1,1,1,0,0,0,1,0]=>2 [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]=>1 [1,0,1,0,1,0,1,0,1,0]=>5 [1,0,1,0,1,0,1,1,0,0]=>4 [1,0,1,0,1,1,0,0,1,0]=>3 [1,0,1,0,1,1,0,1,0,0]=>1 [1,0,1,0,1,1,1,0,0,0]=>3 [1,0,1,1,0,0,1,0,1,0]=>3 [1,0,1,1,0,0,1,1,0,0]=>2 [1,0,1,1,0,1,0,0,1,0]=>1 [1,0,1,1,0,1,0,1,0,0]=>4 [1,0,1,1,0,1,1,0,0,0]=>1 [1,0,1,1,1,0,0,0,1,0]=>2 [1,0,1,1,1,0,0,1,0,0]=>1 [1,0,1,1,1,0,1,0,0,0]=>1 [1,0,1,1,1,1,0,0,0,0]=>2 [1,1,0,0,1,0,1,0,1,0]=>4 [1,1,0,0,1,0,1,1,0,0]=>3 [1,1,0,0,1,1,0,0,1,0]=>2 [1,1,0,0,1,1,0,1,0,0]=>1 [1,1,0,0,1,1,1,0,0,0]=>2 [1,1,0,1,0,0,1,0,1,0]=>1 [1,1,0,1,0,0,1,1,0,0]=>1 [1,1,0,1,0,1,0,0,1,0]=>4 [1,1,0,1,0,1,0,1,0,0]=>3 [1,1,0,1,0,1,1,0,0,0]=>3 [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]=>1 [1,1,0,1,1,1,0,0,0,0]=>1 [1,1,1,0,0,0,1,0,1,0]=>3 [1,1,1,0,0,0,1,1,0,0]=>2 [1,1,1,0,0,1,0,0,1,0]=>1 [1,1,1,0,0,1,0,1,0,0]=>3 [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]=>1 [1,1,1,0,1,1,0,0,0,0]=>1 [1,1,1,1,0,0,0,0,1,0]=>2 [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]=>1 [1,0,1,0,1,0,1,0,1,0,1,0]=>6 [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]=>4 [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]=>4 [1,0,1,0,1,1,0,0,1,0,1,0]=>3 [1,0,1,0,1,1,0,0,1,1,0,0]=>3 [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]=>5 [1,0,1,0,1,1,0,1,1,0,0,0]=>1 [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]=>1 [1,0,1,0,1,1,1,0,1,0,0,0]=>1 [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]=>4 [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]=>2 [1,0,1,1,0,0,1,1,0,1,0,0]=>1 [1,0,1,1,0,0,1,1,1,0,0,0]=>2 [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]=>1 [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]=>4 [1,0,1,1,0,1,0,1,1,0,0,0]=>4 [1,0,1,1,0,1,1,0,0,0,1,0]=>1 [1,0,1,1,0,1,1,0,0,1,0,0]=>1 [1,0,1,1,0,1,1,0,1,0,0,0]=>1 [1,0,1,1,0,1,1,1,0,0,0,0]=>1 [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]=>2 [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]=>3 [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]=>1 [1,0,1,1,1,0,1,0,0,1,0,0]=>1 [1,0,1,1,1,0,1,0,1,0,0,0]=>1 [1,0,1,1,1,0,1,1,0,0,0,0]=>1 [1,0,1,1,1,1,0,0,0,0,1,0]=>2 [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]=>1 [1,0,1,1,1,1,1,0,0,0,0,0]=>2 [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]=>4 [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]=>1 [1,1,0,0,1,0,1,1,1,0,0,0]=>3 [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]=>2 [1,1,0,0,1,1,0,1,0,0,1,0]=>1 [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]=>1 [1,1,0,0,1,1,1,0,0,0,1,0]=>2 [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]=>1 [1,1,0,0,1,1,1,1,0,0,0,0]=>2 [1,1,0,1,0,0,1,0,1,0,1,0]=>1 [1,1,0,1,0,0,1,0,1,1,0,0]=>1 [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]=>1 [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]=>5 [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]=>3 [1,1,0,1,0,1,0,1,1,0,0,0]=>3 [1,1,0,1,0,1,1,0,0,0,1,0]=>3 [1,1,0,1,0,1,1,0,0,1,0,0]=>1 [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]=>1 [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]=>1 [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]=>1 [1,1,0,1,1,0,1,0,0,1,0,0]=>4 [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]=>1 [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]=>1 [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]=>4 [1,1,1,0,0,0,1,0,1,1,0,0]=>3 [1,1,1,0,0,0,1,1,0,0,1,0]=>2 [1,1,1,0,0,0,1,1,0,1,0,0]=>1 [1,1,1,0,0,0,1,1,1,0,0,0]=>2 [1,1,1,0,0,1,0,0,1,0,1,0]=>1 [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]=>4 [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]=>3 [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]=>1 [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]=>1 [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]=>1 [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]=>1 [1,1,1,0,1,0,1,0,0,1,0,0]=>1 [1,1,1,0,1,0,1,0,1,0,0,0]=>3 [1,1,1,0,1,0,1,1,0,0,0,0]=>1 [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]=>1 [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]=>3 [1,1,1,1,0,0,0,0,1,1,0,0]=>2 [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]=>3 [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]=>1 [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]=>1 [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]=>2 [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]=>1
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 $k$-Gorenstein degree of the linear Nakayama algebra corresponding to a Dyck path.
The $k$-Gorenstein degree is the maximal integer $k$ such that the algebra is $k$-Gorenstein. We apply the convention that the value equals the global dimension of the algebra in case the $k$-Gorenstein degree is greater than or equal to the global dimension.
The correspondence between linear Nakayama algebras and Dyck paths is explained on the Nakayama algebras page.
References
[1] Auslander, M., Reiten, I. $k$-Gorenstein algebras and syzygy modules MathSciNet:1259667
Code
gap('LoadPackage("QPA");')

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

InstallMethod(kGortestgiven, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, CoRegA, M, g, i, injA, t, temp, temp2;
    M := LIST[1];
    g := LIST[2]-1;
    A := NakayamaAlgebra(M,GF(3));
    injA := IndecInjectiveModules(A);
    CoRegA := DirectSumOfQPAModules(injA);
    temp := [];
    for i in [1..g] do Append(temp,[Source(ProjectiveCover(NthSyzygy(CoRegA,i)))]);
    od;
    temp2 := [];
    for i in [1..g] do Append(temp2,[i-InjDimensionOfModule(Source(ProjectiveCover(NthSyzygy(CoRegA,i))),30)]);
    od;
    t := Minimum(temp2);
    return(t);
end);

DeclareOperation("kGortestgivenhelp",[IsList]);

InstallMethod(kGortestgivenhelp, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, CoRegA, M, g, i, injA, t, temp, temp2;
    M := LIST[1];
    g := LIST[2]-1;
    A := NakayamaAlgebra(M,GF(3));
    injA := IndecInjectiveModules(A);
    CoRegA := DirectSumOfQPAModules(injA);
    temp := [];
    for i in [1..g] do Append(temp,[Source(ProjectiveCover(NthSyzygy(CoRegA,i)))]);
    od;
    temp2 := [];
    for i in [1..g] do Append(temp2,[i-InjDimensionOfModule(Source(ProjectiveCover(NthSyzygy(CoRegA,i))),30)]);
    od;
    t := Minimum(temp2);
    if (t>=0) then
    return(1);
    else return(0);
    fi;
end);

DeclareOperation("kGordegree",[IsList]);

InstallMethod(kGordegree, "for a representation of a quiver", [IsList],0,function(LIST)
    local A, M, gg, i, temp5, tt;
    M := LIST[1];
    A := NakayamaAlgebra(M,GF(3));
    gg := GlobalDimensionOfAlgebra(A,30);
    temp5 := [];
    for i in [2..gg] do Append(temp5,[kGortestgivenhelp([M,i])]);
    od;
    tt := Sum(temp5)+1;
    return(tt);
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.kGordegree([K]))
Created
Aug 09, 2017 at 11:38 by Rene Marczinzik
Updated
Mar 11, 2026 at 18:03 by Nupur Jain