Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity

We give a new formula for the Littlewood–Richardson coefficients in terms of peelable tableaux compatible with shuffle tableaux, in the same fashion as Remmel–Whitney rule. This gives an efficient way to compute generalized Littlewood–Richardson coefficients for Temperley–Lieb immanants of Jacobi–Trudi matrices. We will also show that our rule behaves well with Bender–Knuth involutions, recovering the symmetry of Littlewood–Richardson coefficients. As an application, we use our rule to prove a special case of a Schur log-concavity conjecture by Lam–Postnikov–Pylyavskyy.