Identifier
Identifier
  • St001015: Dyck paths ⟶ ℤ (values match St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.)
Values
=>
Cc0005;cc-rep
[1,0]=>1 [1,0,1,0]=>1 [1,1,0,0]=>2 [1,0,1,0,1,0]=>0 [1,0,1,1,0,0]=>2 [1,1,0,0,1,0]=>2 [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]=>2 [1,0,1,1,0,1,0,0]=>1 [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]=>3 [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]=>3 [1,1,1,0,0,0,1,0]=>3 [1,1,1,0,0,1,0,0]=>3 [1,1,1,0,1,0,0,0]=>3 [1,1,1,1,0,0,0,0]=>4 [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]=>1 [1,0,1,0,1,1,1,0,0,0]=>2 [1,0,1,1,0,0,1,0,1,0]=>1 [1,0,1,1,0,0,1,1,0,0]=>3 [1,0,1,1,0,1,0,0,1,0]=>1 [1,0,1,1,0,1,0,1,0,0]=>2 [1,0,1,1,0,1,1,0,0,0]=>2 [1,0,1,1,1,0,0,0,1,0]=>3 [1,0,1,1,1,0,0,1,0,0]=>3 [1,0,1,1,1,0,1,0,0,0]=>2 [1,0,1,1,1,1,0,0,0,0]=>4 [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]=>3 [1,1,0,0,1,1,0,1,0,0]=>2 [1,1,0,0,1,1,1,0,0,0]=>4 [1,1,0,1,0,0,1,0,1,0]=>1 [1,1,0,1,0,0,1,1,0,0]=>2 [1,1,0,1,0,1,0,0,1,0]=>1 [1,1,0,1,0,1,0,1,0,0]=>0 [1,1,0,1,0,1,1,0,0,0]=>3 [1,1,0,1,1,0,0,0,1,0]=>3 [1,1,0,1,1,0,0,1,0,0]=>2 [1,1,0,1,1,0,1,0,0,0]=>3 [1,1,0,1,1,1,0,0,0,0]=>4 [1,1,1,0,0,0,1,0,1,0]=>2 [1,1,1,0,0,0,1,1,0,0]=>4 [1,1,1,0,0,1,0,0,1,0]=>2 [1,1,1,0,0,1,0,1,0,0]=>3 [1,1,1,0,0,1,1,0,0,0]=>4 [1,1,1,0,1,0,0,0,1,0]=>2 [1,1,1,0,1,0,0,1,0,0]=>3 [1,1,1,0,1,0,1,0,0,0]=>3 [1,1,1,0,1,1,0,0,0,0]=>4 [1,1,1,1,0,0,0,0,1,0]=>4 [1,1,1,1,0,0,0,1,0,0]=>4 [1,1,1,1,0,0,1,0,0,0]=>4 [1,1,1,1,0,1,0,0,0,0]=>4 [1,1,1,1,1,0,0,0,0,0]=>5 [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]=>1 [1,0,1,0,1,0,1,1,1,0,0,0]=>2 [1,0,1,0,1,1,0,0,1,0,1,0]=>0 [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]=>1 [1,0,1,0,1,1,0,1,0,1,0,0]=>2 [1,0,1,0,1,1,0,1,1,0,0,0]=>2 [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]=>2 [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]=>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]=>3 [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]=>4 [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]=>2 [1,0,1,1,0,1,0,1,0,0,1,0]=>1 [1,0,1,1,0,1,0,1,0,1,0,0]=>0 [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]=>3 [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]=>4 [1,0,1,1,1,0,0,1,0,0,1,0]=>2 [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]=>4 [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]=>3 [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]=>3 [1,0,1,1,1,1,0,0,0,0,1,0]=>4 [1,0,1,1,1,1,0,0,0,1,0,0]=>4 [1,0,1,1,1,1,0,0,1,0,0,0]=>4 [1,0,1,1,1,1,0,1,0,0,0,0]=>3 [1,0,1,1,1,1,1,0,0,0,0,0]=>5 [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]=>2 [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]=>2 [1,1,0,0,1,1,0,0,1,1,0,0]=>4 [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]=>3 [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]=>4 [1,1,0,0,1,1,1,0,1,0,0,0]=>3 [1,1,0,0,1,1,1,1,0,0,0,0]=>5 [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]=>2 [1,1,0,1,0,0,1,1,0,0,1,0]=>2 [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]=>3 [1,1,0,1,0,1,0,0,1,0,1,0]=>1 [1,1,0,1,0,1,0,0,1,1,0,0]=>2 [1,1,0,1,0,1,0,1,0,0,1,0]=>0 [1,1,0,1,0,1,0,1,0,1,0,0]=>0 [1,1,0,1,0,1,0,1,1,0,0,0]=>1 [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]=>2 [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]=>4 [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]=>4 [1,1,0,1,1,0,0,1,0,0,1,0]=>2 [1,1,0,1,1,0,0,1,0,1,0,0]=>3 [1,1,0,1,1,0,0,1,1,0,0,0]=>3 [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]=>1 [1,1,0,1,1,0,1,1,0,0,0,0]=>4 [1,1,0,1,1,1,0,0,0,0,1,0]=>4 [1,1,0,1,1,1,0,0,0,1,0,0]=>4 [1,1,0,1,1,1,0,0,1,0,0,0]=>3 [1,1,0,1,1,1,0,1,0,0,0,0]=>4 [1,1,0,1,1,1,1,0,0,0,0,0]=>5 [1,1,1,0,0,0,1,0,1,0,1,0]=>2 [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]=>4 [1,1,1,0,0,0,1,1,0,1,0,0]=>3 [1,1,1,0,0,0,1,1,1,0,0,0]=>5 [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]=>3 [1,1,1,0,0,1,0,1,0,0,1,0]=>2 [1,1,1,0,0,1,0,1,0,1,0,0]=>1 [1,1,1,0,0,1,0,1,1,0,0,0]=>4 [1,1,1,0,0,1,1,0,0,0,1,0]=>4 [1,1,1,0,0,1,1,0,0,1,0,0]=>3 [1,1,1,0,0,1,1,0,1,0,0,0]=>4 [1,1,1,0,0,1,1,1,0,0,0,0]=>5 [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]=>3 [1,1,1,0,1,0,0,1,0,0,1,0]=>2 [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]=>4 [1,1,1,0,1,0,1,0,0,0,1,0]=>2 [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]=>4 [1,1,1,0,1,1,0,0,0,0,1,0]=>4 [1,1,1,0,1,1,0,0,0,1,0,0]=>3 [1,1,1,0,1,1,0,0,1,0,0,0]=>4 [1,1,1,0,1,1,0,1,0,0,0,0]=>4 [1,1,1,0,1,1,1,0,0,0,0,0]=>5 [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]=>5 [1,1,1,1,0,0,0,1,0,0,1,0]=>3 [1,1,1,1,0,0,0,1,0,1,0,0]=>4 [1,1,1,1,0,0,0,1,1,0,0,0]=>5 [1,1,1,1,0,0,1,0,0,0,1,0]=>3 [1,1,1,1,0,0,1,0,0,1,0,0]=>4 [1,1,1,1,0,0,1,0,1,0,0,0]=>4 [1,1,1,1,0,0,1,1,0,0,0,0]=>5 [1,1,1,1,0,1,0,0,0,0,1,0]=>3 [1,1,1,1,0,1,0,0,0,1,0,0]=>4 [1,1,1,1,0,1,0,0,1,0,0,0]=>4 [1,1,1,1,0,1,0,1,0,0,0,0]=>4 [1,1,1,1,0,1,1,0,0,0,0,0]=>5 [1,1,1,1,1,0,0,0,0,0,1,0]=>5 [1,1,1,1,1,0,0,0,0,1,0,0]=>5 [1,1,1,1,1,0,0,0,1,0,0,0]=>5 [1,1,1,1,1,0,0,1,0,0,0,0]=>5 [1,1,1,1,1,0,1,0,0,0,0,0]=>5 [1,1,1,1,1,1,0,0,0,0,0,0]=>6
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Description
Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path.
Code
gap('LoadPackage("QPA");')

gap.eval('DeclareOperation("statistic", [IsList]);')

gap_code = r"""
InstallMethod(statistic, "for a representation of a quiver", [IsList], 0, function(L)
local list, A, R, RR;
list := L;
A := NakayamaAlgebra(GF(3), list);
R := IndecInjectiveModules(A);
RR := Filtered(R,x->DominantDimensionOfModule(DualOfModule(x),1) = 1);
return(Size(RR));
end);
"""

gap.eval(gap_code)

def kupisch(D):
    """
    sage: [kupisch(D) for D in DyckWords(3)]
    [[2, 2, 2, 1], [2, 3, 2, 1], [3, 2, 2, 1], [3, 3, 2, 1], [4, 3, 2, 1]]
    """
    H = D.heights()
    return [1+H[i] for i, s in enumerate(D) if s == 0]+[1]

def statistic(D):
    return gap.statistic(kupisch(D.reverse()))

# CodeLanguage: GAP4 with QPA
LoadPackage("QPA");
DeclareOperation("statistic", [IsList]);

InstallMethod(statistic, "for a representation of a quiver", [IsList], 0, function(L)
local list, A, R, RR;
list := L;
A := NakayamaAlgebra(GF(3), list);
R := IndecInjectiveModules(A);
RR := Filtered(R,x->DominantDimensionOfModule(DualOfModule(x),1) = 1);
return(Size(RR));
end);
Created
Oct 29, 2017 at 17:49 by Rene Marczinzik
Updated
Apr 25, 2018 at 13:13 by Martin Rubey