It is not difficult to show that the number of conjugacy classes in the alternating group $A_n$ is given by

classes in the alternating group = no. of even partitions + no. of self-transpose partitions

Note that a partition is *even* if it is the cycle decomposition of an even permutation.

Likewise, using some Clifford theory and the representation theory of $S_n$, one can show that the number of irreducible representations of $A_n$ is given by

irreps. of $A_n$ = $\frac 12$(no. of non-self transpose partitions) + no. of self-transpose partitions

Equating these leads to the identity:

$2\times $ no. of even partitions - no. of self-transpose partitions = no. of partitions

In his book *Representations of Finite Groups*, Musili refers to this as a *bizarre identity*.

Question.Is there a proof of this identity which does not use the representation theory of alternating groups? Better still, is there a bijective proof?

Enumerative Combinatorics, vol. 1, 2nd ed., and the ends of Sections 7.7 and 7.14 of vol. 2. $\endgroup$ – Richard Stanley Oct 23 '12 at 15:27