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



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



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> irreps. of $A\_n$ = $\frac 12$(no. of non-self transpose partitions) + no. of self-transpose partitions
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Equating these leads to the identity:



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> $2\times $ no. of even partitions - no. of self-transpose partitions = no. of partitions
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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?
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Identity involving partitions coming from representations of alternating groups
This question got answered by Gjergji Zaimi and Richard Stanley in the comments. I simply reproduce their comments here as an answer:


A very simple explanation for this identity comes from the theory of symmetric functions. The ring $\Lambda$ of symmetric functions in infinitely many variables comes with an involution $\omega$, which interchanges the complete symmetric function $h\_\lambda$ with the elementary symmetric function $e\_\lambda$ for each partition $\lambda$.


Comparing the answers obtained for the trace of $\omega$ on homogeneous symmetric functions of degree $n$ using Schur functions and power sum symmetric functions yields the identity in question, for $\omega(s\_\lambda)=s\_{\lambda'}$ (giving trace as the number of self-transpose partitions) and $\omega(p\_\lambda)=\epsilon(\lambda)p\_\lambda$, where $\epsilon(\lambda)$ is the sign of a permutation with cycle decomposition $\lambda$ (giving trace as number of even partitions minus number of odd partitions).


Proofs of these facts concerning symmetric functions can be found in Stanley's *Enumerative Combinatorics 2* (Sections 7.7 and 7.14).


A bijective proof for this identity was given by Marc van Leeuwen to the same question on <https://math.stackexchange.com/a/102293/10126>; he constructs an explicit bijection between the sets of even and odd partitions which do not have distinct odd parts. Also, there is a fairly standard bijection between partitions with distinct odd parts and self-transpose partitions.


