Computing the Number of Types of Infinite Length

We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of 1-types and the length of the sequences. Specifically, if $\kappa \le \lambda$, then $$\underset{\mathrm{\mid }A\mathrm{\mid }=\lambda }{\sup }\mathrm{\mid }{S}^{\kappa }(A)\mathrm{\mid }=(\underset{\mathrm{\mid }A\mathrm{\mid }=\lambda }{\sup }\mathrm{\mid }{S}^1(A)\mathrm{\mid }{)}^{\kappa }$$ We show that this holds for any abstract elementary class with $\lambda$ amalgamation, but it is new for first order theories when $\kappa$ is infinite. No such calculation is possible for nonalgebraic types. We introduce a generalization of nonalgebraic types for which the same upper bound holds.