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We looked into this with Greta and Damir in On the largest Kronecker and Littlewood–Richardson coefficients. Unfortunately, other than the Harris–Willenbring identity we didn't find much. For our bounds we even had to invent a new (not too difficult) LR-identity which follows from the skew Cauchy identity (it's Lemma 5.1 in the paper): $$ \sum_{\lambda\vdash n} \left( c^{\lambda}_{\mu\nu}\right)^2 \, = \, \sum_{\alpha, \beta, \gamma, \delta} c^{\mu}_{\alpha\gamma} c^{\mu}_{\alpha \delta} c^{\nu}_{\beta\gamma} c^{\nu}_{\beta \delta}. $$