Exploiting Probabilistic Independence for Permutations

The multiplicities in the decomposition (Equation 0.1) are equivalent to the famousLittlewood-Richardson coe cients, 1 and in this appendix, we describe a result known as the Littlewood-Richardson (LR) rule which will allow us to compute the Littlewood-Richardson coe cients tractably (at least for low-order terms). There are several methods for computing these numbers (see [Knutson and Tao, 1999, Vakil, 2006], for example) but it is known ([Narayanan, 2006]) that, in general, the problem of computing the LittlewoodRichardson coe cients is #P -hard in general, and as we will see, involves enumerating the integer feasible points of a linearly constrained polytope.