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]. 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