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We present an algorithm that computes the composition factors of the n-thtensor power of the free associative algebra on a vector space. The compositionfactors admit a description in terms of certain coefficients $c_{\lambda\mu}$determining their irreducible structure. By reinterpreting these coefficientsas counting the number of ways to solve certain `decomposition-puzzles' we areable to design an efficient algorithm extending the range of computation by afactor of over 750. Furthermore, by visualising the data appropriately, we gaininsights into the nature of the coefficients leading to the development of anew representation theoretic framework called PD-modules.

An efficient algorithm computing composition factors of $T(V)^{\otimes n}$