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Here's an example I found by searching "combinatorial proof" in Stanley's supplemental problems to EC2 Chapter 7: see here for the problems and here for the solutions.

If you look at the solution to problem 106(b) there, you will see Stanley notes that it implies $$ \#\{(u,v)\in S_n\times S_n\colon uv = vu\} = \#\{(u,v)\in S_n \times S_n \colon u^2=v^2 \}.$$ Here $S_n$ is the symmetric group on $n$ letters. The proof is via some symmetric function identities and Stanley remarks that the same is true for any finite group $G$ all of whose complex representations are equivalent to real representations.

He asks there if there is a combinatorial proof of this identity. I have no idea if this is easy.

Here's an example I found by searching "combinatorial proof" in Stanley's supplemental problems to EC2 Chapter 7: see here for the problems and here for the solutions.

If you look at the solution to problem 106(b) there, you will see Stanley notes that it implies $$ \#\{(u,v)\in S_n\times S_n\colon uv = vu\} = \#\{(u,v)\in S_n \times S_n \colon u^2=v^2 \}.$$ Here $S_n$ is the symmetric group on $n$ letters. The proof is via some symmetric function identities and Stanley remarks that the same is true for any finite group $G$ all of whose complex representations are equivalent to real representations.

He asks there if there is a combinatorial proof of this identity. I have no idea if this is easy.

EDIT: This is Exercise 108(b) in the Supplementary Problems of Chapter 7 as in the published version of the 2nd edition of EC2.

Here's an example I found by searching "combinatorial proof" in Stanley's supplemental problems to EC2 Chapter 7: see here for the problems and here for the solutions.

If you look at the solution to problem 106(b) there, you will see Stanley notes that it implies $$ \#\{(u,v)\in S_n\times S_n\colon uv = vu\} = \#\{(u,v)\in S_n \times S_n \colon u^2=v^2 \}.$$ Here $S_n$ is the symmetric group on $n$ letters. The proof is via some symmetric function identities and Stanley remarks that the same is true for any finite group $G$ all of whose complex representations are equivalent to real representations.

He asks there if there is a combinatorial proof of this identity. I have no idea if this is easy.

Here's an example I found by searching "combinatorial proof" in Stanley's supplemental problems to EC2 Chapter 7: see here for the problems and here for the solutions.

If you look at the solution to problem 106(b) there, you will see Stanley notes that it implies $$ \#\{(u,v)\in S_n\times S_n\colon uv = vu\} = \#\{(u,v)\in S_n \times S_n \colon u^2=v^2 \}.$$ Here $S_n$ is the symmetric group on $n$ letters. The proof is via some symmetric function identities and Stanley remarks that the same is true for any finite group $G$ all of whose complex representations are equivalent to real representations.

He asks there if there is a combinatorial proof of this identity. I have no idea if this is easy.

Here's an example I found by searching "combinatorial proof" in Stanley's supplemental problems to EC2 Chapter 7: see here for the problems and here for the solutions.

If you look at the solution to problem 106(b) there, you will see Stanley notes that it implies $$ \#\{(u,v)\in S_n\times S_n\colon uv = vu\} = \#\{(u,v)\in S_n \times S_n \colon u^2=v^2 \}.$$ Here $S_n$ is the symmetric group on $n$ letters. The proof is via some symmetric function identities and Stanley remarks that the same is true for any finite group $G$ all of whose complex representations are equivalent to real representations.

He asks there if there is a combinatorial proof of this identity. I have no idea if this is easy.

Here's an example I found by searching "combinatorial proof" in Stanley's supplemental problems to EC2 Chapter 7: see here for the problems and here for the solutions.

If you look at the solution to problem 106(b) there, you will see Stanley notes that it implies $$ \#\{(u,v)\in S_n\times S_n\colon uv = vu\} = \#\{(u,v)\in S_n \times S_n \colon u^2=v^2 \}.$$ Here $S_n$ is the symmetric group on $n$ letters. The proof is via some symmetric function identities and Stanley remarks that the same is true for any finite group $G$ all of whose complex representations are equivalent to real representations.

He asks there if there is a combinatorial proof of this identity. I have no idea if this is easy.