There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the Knutson-Tao hive model rule and many more. Any of these rules provides an effective algorithm for computing products of Schur functions. It is known that computing Littlewood-Richardson coefficients is, in general, #P complete.

I am interested in the most computationially efficient Littlewood-Richardson rule. Since any rule will be in #P, for which I am not aware of easy characterizations of run time (please correct me if I'm wrong on this), I suspect any answer to this question would have to be based on experimental evidence. No doubt the various computer algebra systems have coded the various rules, computed a bunch of examples and checked which one was fastest. In one of my graduate courses, we were told the Remmel rule was the easiest, though this was for by hand computations of small examples. At the end of Exercise 2.7.10 of book *Symmetric Functions, Schubert Polynomials and Degeneracy Loci*, Manivel asserts that the Lascoux-Schutzenberger transition equations (see here) for Schubert polynomials (and hence Stanley symmetric functions) provides the most computationally efficient means of computing the product of Schur functions.

1) What is the (experimentally) most efficient method of computing a single Littlewood-Richardson coefficient?

2) Can anyone confirm or refute Manivel's assertion that the Lascoux-Schutzenberger transition equations are the most efficient rule for computing the product of Schur functions?

allthe terms in the expansion of $s_\lambda s_\mu$, one method is to work with a sufficiently large but finite number of variables, express the product as a linear combination of Schur functions with unknown coefficients, write the Schur functions as bialternants (quotients of determinants), and specialize the variables to real numbers sufficiently generically to be able solve the resulting system of linear equations for the unknown coefficients. I believe that John Stembridge uses this technique for some of his SF computations. $\endgroup$