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 symmetric 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 symmetric 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?

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?

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 symmetric 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 symmetric partitions

Equating these leads to the identity:

$2\times $ no. of even partitions - no. of symmetric 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?

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 symmetric 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 symmetric 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?