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][1] in [Sums of squares of Littlewood–Richardson coefficients and $\operatorname{GL}_n$-harmonic polynomials](https://doi.org/10.1007/978-1-4939-1590-3_11). 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?**


  [1]: https://www.symmetricfunctions.com/littlewoodRichardson.htm#harrisWillenbring