  We determine the structure of the partition algebra $P_n(Q)$ (a generalized
Temperley-Lieb algebra) for specific values of $Q \in \C$, focusing on the
quotient which gives rise to the partition function of $n$ site $Q$-state Potts
models (in the continuous $Q$ formulation) in arbitrarily high lattice
dimensions (the mean field case). The algebra is non-semi-simple iff $Q$ is a
non-negative integer less than $n$. We determine the dimension of the key
irreducible representation in every specialization.
