Let $\Lambda$ be the ring of symmetric functions in infinitely many variables, $x_1$, $x_2$, .... For $f \in \Lambda$, let $\Delta(f) \in \Lambda \otimes \Lambda$ be $f(x_1 \otimes 1, 1 \otimes x_1, x_2 \otimes 1, 1 \otimes x_2, \ldots)$. It turns out that $\Delta$ of a Schur function is given by Littlewood-Richardson coefficients: $\Delta(s_{\nu}) = \sum_{\lambda, \mu} c_{\lambda \mu}^{\nu} s_{\lambda} \otimes s_{\mu}$. Littlewood-Richardson coefficients also describe multiplication: $s_{\lambda} \cdot s_{\mu} = \sum c_{\lambda \mu}^{\nu} s_{\nu}$. And we have the identity (elementary once you unravel what it means) $\Delta(fg) = \Delta(f) \Delta(g)$.

So, for any partitions $\kappa$, $\lambda$, $\mu$, $\nu$, we must have $$\sum_{\rho} c_{\kappa \lambda}^{\rho} c_{\mu \nu}^{\rho} = \sum_{\sigma_1, \sigma_2, \sigma_3, \sigma_4} c_{\sigma_1 \sigma_2}^{\kappa} c_{\sigma_2 \sigma_3}^{\lambda} c_{\sigma_3 \sigma_4}^{\mu} c_{\sigma_4 \sigma_1}^{\nu}.$$

I'm curious whether anyone knows a bijective proof of this? No motivation right now, but I find it pretty useful to know which algebraic facts have good combinatorial explanations.