I am currently reading William Fulton's Young Tableaux and struggling to understand the proof of Corollary 2 in Section 5 of the book.
Suppose that $\lambda$ and $\mu$ are Young diagrams (or partitions) and let $\tilde{\lambda}$ and $\tilde{\mu}$ be their conjugates. Assume that $V_0$ is a standard tableau of some shape $\nu$ and let $\tilde{V_0}$ be the conjugate tableau.
Denote by $\mathcal{T}(\lambda,\mu,V_0)$ the number of pairs $(T,U)$, where $T$ is a tableau of the shape $\lambda$, $U$ is a tableau of the shape $\mu$ and $T\cdot U=V_0$ (we use the defined earlier product of tableaux).
It seems to me that Fulton uses the fact that $\mathcal{T}(\lambda,\mu,V_0)=\mathcal{T}(\tilde{\lambda},\tilde{\mu},\tilde{V_0})$ but I don't understand why this equality holds. Perhaps, there is an "obvious" bijection but I can't see it (maybe one can use the Knuth equivalence somehow).
Note that this corollary is at the beginning of the chapter about the Littlewood-Richardson rule, so we can't yet use the properties of Littlewood-Richardson numbers $c_{\lambda\mu}^{\nu}$.