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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.

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    $\begingroup$ Your coefficient has both $P$ and $Q$ with either one row or one column. If either holds then the Littlewood–Richardson Rule can be replaced with the easier Pieri's Rule. (Sometimes called Young's rule for the single part case, i.e. $C^R_{P (q)}$.) I often see the LR rule applied in the literature when this easier result suffices. $\endgroup$ – Mark Wildon Nov 19 '19 at 18:00
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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.

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