I am not aware of that many identities that involve several Littlewood–Richardson coefficients. One recent identity, is a generating function as sum of squares of LR-coefficients, due to Harris and Willenbring in Sums of squares of Littlewood–Richardson coefficients and $\operatorname{GL}_n$-harmonic polynomials. This identity is really interesting.

There is also a relation between two sums of LR-coefficients, due to Coquereaux and Zuber, but I find this less interesting for this question.

So my question is, are there some interesting relations involving (potentially weighted) sums of LR-coefficients?