Finding Littlewood-Richardson coefficients without using identities The Littlewood-Richardson coefficients $C^{R}_{QP}$ for some partitions $R, Q, P$ can usually be dealt with using identities like for example $$C^{R}_{QP} = 0 \quad \text{ if } \quad|R| \neq  |Q| + |P|$$ or $$C^{R}_{Q\bullet} = \delta^{R}_{Q}.$$ However, how does one actually calculate something like $$C^{(2,1)}_{(1), (1,1)}?$$
Since $|R| = |Q| + |P|$ and $P \neq \bullet$ I cannot use the above identities and I struggle to find any method of actually calculating the coefficients.
 A: There are several approaches.
Use linear algebra. Compute Schur polynomials (using the Jacobi-Trudi identity, say), and then use the fact that the coefficient of $s_\nu$ in the product $s_\lambda s_\mu$ is a Littlewood-Richardson coefficient.
Use a combinatorial interpretation. 
One can count so called Littlewood-Richardson tableaux, or lattice points in so-called Berenstein-Zelevinsky polytopes. The latter can be done rather efficient using some lattice-point counting program, such as lattE.
However, note that it has been proved that computing Littewood-Richardson coefficients is #$P$-complete, meaning that there is no super-efficient algorithm for computing these (that is, no nice closed formula).
If you want to compute several coefficients, a recursive approach might be suitable. One can show that the multiplicative constants for so called shifted Schur functions are generalizations of Littlewood-Richardson coefficients, and these do satisfy a very nice recursive formula.
See the paper and proposition 3.4 in 
A Littlewood-Richardson Rule for Factorial Schur Functions
by Alexander I. Molev and Bruce E. Sagan.
See also this answer.
