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PeterBurbery`Combinatorics`
SelfConjugatePartitionQ  |
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Examples
(5)
Basic Examples
(2)
A partition of 10 and its conjugate:
In[1]:=
p={6,3,1};cp=
[p]
 | ConjugatePartition |
Out[1]=
{3,2,2,1,1,1}
This partition is not self-conjugate because it doesn't equal its conjugate.
In[2]:=
p===cp
Out[2]=
False
Use the function FerrersDiagram to show the Ferrers diagrams of the partition and its conjugate together:
In[3]:=
[p],
[cp]//Row[#,Spacer[10]]&
 | FerrersDiagram |
| FerrersDiagram |
Out[3]=
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Some partitions are self-conjugate:
In[1]:=
s={5,2,1,1,1};
In[2]:=
 | ConjugatePartition |
Out[2]=
{5,2,1,1,1}
In[3]:=
s===
[s]
| ConjugatePartition |
Out[3]=
True
The partition is self-conjugate:
In[4]:=
| SelfConjugatePartitionQ |
Out[4]=
True
Find the total:
In[5]:=
Total[s]
Out[5]=
10
Find all integer partitions of 10:
In[6]:=
IntegerPartitions[10]
Out[6]=
{{10},{9,1},{8,2},{8,1,1},{7,3},{7,2,1},{7,1,1,1},{6,4},{6,3,1},{6,2,2},{6,2,1,1},{6,1,1,1,1},{5,5},{5,4,1},{5,3,2},{5,3,1,1},{5,2,2,1},{5,2,1,1,1},{5,1,1,1,1,1},{4,4,2},{4,4,1,1},{4,3,3},{4,3,2,1},{4,3,1,1,1},{4,2,2,2},{4,2,2,1,1},{4,2,1,1,1,1},{4,1,1,1,1,1,1},{3,3,3,1},{3,3,2,2},{3,3,2,1,1},{3,3,1,1,1,1},{3,2,2,2,1},{3,2,2,1,1,1},{3,2,1,1,1,1,1},{3,1,1,1,1,1,1,1},{2,2,2,2,2},{2,2,2,2,1,1},{2,2,2,1,1,1,1},{2,2,1,1,1,1,1,1},{2,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1}}
Select self conjugate partitions:
In[7]:=
Select
[IntegerPartitions[10]]
| SelfConjugatePartitionQ |
Out[7]=
{{5,2,1,1,1},{4,3,2,1}}
Using the function FerrersDiagram, verify that a self-conjugate partition has a symmetric Ferrers diagram:
In[8]:=
| FerrersDiagram |
Out[8]=
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Verify that all the partitions selected are self-conjugate.
In[9]:=
AllTrue
Select
[IntegerPartitions[10]]
| SelfConjugatePartitionQ |
 | SelfConjugatePartitionQ |
Out[9]=
True
Visualize them with Ferrers diagrams:
In[10]:=
| FerrersDiagram |
| SelfConjugatePartitionQ |
Out[10]=
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Applications
(3)
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