Self-Conjugate Partition

Author

Eric W. Weisstein
December 24, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/Self-ConjugatePartition.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Self-ConjugatePartition.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

ConjugatePartition[l_List]:=Module[{i,r=Reverse[l],n=Length[l]},
Table[n+1-Position[r,_?(#>=i&),Infinity,1][[1,1]],{i,l[[1]]}]
]

SelfConjugateQ[l_List]:=(l==ConjugatePartition[l])

Table

Table[Select[Partitions[n],SelfConjugateQ],{n,10}]//ColumnForm

Table[Count[Partitions[n],_?SelfConjugateQ],{n,25}]//Timing

Table[Count[Partitions[n],_?SelfConjugateQ],{n,30}]//Timing

Product

(∏_(k=0)^∞)((x^(2 k+1))+1)==(QPochhammer2[-x, (x^2)])

Product[1+x^(2k+1),{k,0,∞}]

FullSimplify[%]

Series[QPochhammer[-x,(x^2)],{x,0,10}]

(∏_k^∞)(1/((x^k)+1))==(2/(QPochhammer2[-1, x]))

Product[1/(1+x^k),{k,∞}]

FullSimplify[%]

Series[(2/(QPochhammer[-1,x])),{x,0,10}]

Graphics

sc[n_]:=sc[n]=Select[Partitions[n],SelfConjugateQ]
Show[Graphics[{PointSize[.05],
Table[FerrersDiagram[sc[n][[k]]][[1]] /.{PointSize[.05]->{},Point[l_List]:>Point[l+{10(k-1),-2(n-3)-Sum[sc[i][[1,1]],{i,3,n}]-If[Length[sc[n]]==1,0,sc[n][[k,1]]-Max[First/@sc[n]]]}]},
{n,3,10},{k,Length[sc[n]]}
]//Flatten},AspectRatio->Automatic]]

Show[Graphics[{PointSize[.02],
Table[FerrersDiagram[sc[n][[k]]][[1]] /.{PointSize[.05]->{},Point[l_List]:>Point[l+{10(k-1),-2(n-16)-If[Length[sc[n]]==1,0,sc[n][[k,1]]-Max[First/@sc[n]]]}]},
{n,16,16},{k,Length[sc[n]]}
]//Flatten},AspectRatio->Automatic],PlotRange->All]