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Identifier

St001283: Integer partitions ⟶ ℤ

Values

[1] => 1

[2] => 0

[1,1] => 1

[3] => 0

[2,1] => 0

[1,1,1] => 1

[4] => 0

[3,1] => 0

[2,2] => 0

[2,1,1] => 1

[1,1,1,1] => 2

[5] => 0

[4,1] => 0

[3,2] => 0

[3,1,1] => 0

[2,2,1] => 0

[2,1,1,1] => 0

[1,1,1,1,1] => 1

[6] => 0

[5,1] => 0

[4,2] => 0

[4,1,1] => 0

[3,3] => 0

[3,2,1] => 0

[3,1,1,1] => 1

[2,2,2] => 0

[2,2,1,1] => 1

[2,1,1,1,1] => 2

[1,1,1,1,1,1] => 1

[7] => 0

[6,1] => 0

[5,2] => 0

[5,1,1] => 0

[4,3] => 0

[4,2,1] => 0

[4,1,1,1] => 0

[3,3,1] => 0

[3,2,2] => 0

[3,2,1,1] => 0

[3,1,1,1,1] => 0

[2,2,2,1] => 0

[2,2,1,1,1] => 0

[2,1,1,1,1,1] => 0

[1,1,1,1,1,1,1] => 1

[8] => 0

[7,1] => 0

[6,2] => 0

[6,1,1] => 0

[5,3] => 0

[5,2,1] => 0

[5,1,1,1] => 0

[4,4] => 0

[4,3,1] => 0

[4,2,2] => 0

[4,2,1,1] => 0

[4,1,1,1,1] => 1

[3,3,2] => 0

[3,3,1,1] => 0

[3,2,2,1] => 0

[3,2,1,1,1] => 0

[3,1,1,1,1,1] => 0

[2,2,2,2] => 0

[2,2,2,1,1] => 1

[2,2,1,1,1,1] => 2

[2,1,1,1,1,1,1] => 0

[1,1,1,1,1,1,1,1] => 3

[9] => 0

[8,1] => 0

[7,2] => 0

[7,1,1] => 0

[6,3] => 0

[6,2,1] => 0

[6,1,1,1] => 0

[5,4] => 0

[5,3,1] => 0

[5,2,2] => 0

[5,2,1,1] => 0

[5,1,1,1,1] => 0

[4,4,1] => 0

[4,3,2] => 0

[4,3,1,1] => 0

[4,2,2,1] => 0

[4,2,1,1,1] => 0

[4,1,1,1,1,1] => 0

[3,3,3] => 0

[3,3,2,1] => 0

[3,3,1,1,1] => 1

[3,2,2,2] => 0

[3,2,2,1,1] => 0

[3,2,1,1,1,1] => 0

[3,1,1,1,1,1,1] => 0

[2,2,2,2,1] => 0

[2,2,2,1,1,1] => 0

[2,2,1,1,1,1,1] => 0

[2,1,1,1,1,1,1,1] => 0

[1,1,1,1,1,1,1,1,1] => 2

[10] => 0

[9,1] => 0

[8,2] => 0

[8,1,1] => 0

[7,3] => 0

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Description

The number of finite solvable groups that are realised by the given partition over the complex numbers.
A finite group $G$ is realised by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.

Code

@cached_function
def small_solvable_group_degrees(n):
    return [(G, Partition(sorted([ZZ(e) for e,f in G.CharacterDegrees() for i in range(ZZ(f))], reverse=True)))
            for G in gap.AllSmallGroups(n) if G.IsSolvableGroup().sage()]

def statistic(la):
    n = sum(e^2 for e in la)
    return sum(1 for G, mu in small_solvable_group_degrees(n) if mu == la)

#CodeLanguage: gap

DeclareOperation("partitiontranslate1",[IsList]);

InstallMethod(partitiontranslate1, "for a representation of a quiver", [IsList],0,function(LIST)

local L,T,WW;

L:=LIST[1];
T:=LIST[2];
WW:=Filtered(L,x->(x=T)=true);
return([T,Size(WW)]);

end);


DeclareOperation("partitiontranslate2",[IsList]);

InstallMethod(partitiontranslate2, "for a representation of a quiver", [IsList],0,function(LIST)

local L,WW2,WW3;

L:=LIST[1];
WW2:=[];for i in [1..Sum(L)] do Append(WW2,[partitiontranslate1([L,i])]);od;
WW3:=Filtered(WW2,x->x[2]>0);
return(WW3);

end);

DeclareOperation("partitiongroup",[IsList]);

InstallMethod(partitiongroup, "for a representation of a quiver", [IsList],0,function(LIST)

local L,U,s,GG,W,GG2;

L:=LIST[1];

U:=[];for i in L do Append(U,[i^2]);od;
s:=Sum(U);
GG:=AllSmallGroups(s);
W:=Filtered(GG,x->CharacterDegrees(x)=partitiontranslate2([L]));
return(Size(W));

end);


DeclareOperation("partitiongroupsolvable",[IsList]);

InstallMethod(partitiongroupsolvable, "for a representation of a quiver", [IsList],0,function(LIST)

local L,U,s,GG,W,GG2;

L:=LIST[1];

U:=[];for i in L do Append(U,[i^2]);od;
s:=Sum(U);
GG:=AllSmallGroups(s);
GG2:=Filtered(GG,x->IsSolvableGroup(x)=true);
W:=Filtered(GG2,x->CharacterDegrees(x)=partitiontranslate2([L]));
return(Size(W));

end);

Created

Nov 04, 2018 at 00:57 by Rene Marczinzik

Updated

Nov 04, 2018 at 20:33 by Martin Rubey