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IsSelfConjugate on IntegerPartition ​

1 when λ equals its conjugate, 0 otherwise.

On IntegerPartition

Called as CombinatorialStat(x, "IsSelfConjugate"). The head's page has its examples.

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FindStat St001122

The multiplicity of the sign representation in the Kronecker square corresponding to a partition.

ObjectValue
[1]1
[2]0
[3]0
[4]0
[5]0
[6]0
[7]0
[8]0

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001122 has the rest.