Question

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?

Answer 1

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 http://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.