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Community-contributed installable additions to the Wolfram Language
Wolfram`TensorNetworks`Symmetry`
TransposePartition  |
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Each entry of the result is the count of partition parts at least as large as the column index: the j-th element of the conjugate is .
Count[partition,x_/;x≥j]
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TransposePartition
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TransposePartition
TransposePartition[TransposePartition[par]]
par
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Self-conjugate partitions (those whose Young diagram is symmetric about its main diagonal) satisfy . Examples include and .
TransposePartition[par]===par
{3,2,1}
{4,2,1,1}
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The length of the conjugate equals the largest part of the input: equals .
Length[TransposePartition[par]]
First[par]
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TransposePartition[{}]
{}
PartitionQ[{}]
False
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Non-partition input (anything failing ) issues a message and returns .
PartitionQ[…]
TransposePartition::notpar
$Failed
Examples
(20)
Basic Examples
(4)
The conjugate of the partition {4, 2, 1}:
In[1]:=
 | TransposePartition |
Out[1]=
{3,2,1,1}
TransposePartition is a self-inverse operation; applying it twice returns the original partition:
In[1]:=
| TransposePartition |
 | TransposePartition |
Out[1]=
{4,2,1}
The partition {3, 2, 1} is self-conjugate: its Young diagram is symmetric about the main diagonal, so the result equals the input:
In[1]:=
| TransposePartition |
Out[1]=
{3,2,1}
For any self-conjugate partition the self-inverse property is doubly demonstrated:
In[1]:=
| TransposePartition |
| TransposePartition |
Out[1]=
{3,2,1}
Scope
(9)
Applications
(3)
Properties & Relations
(4)
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