 Type class

The type class $T_n(q)$ contains all length-$n$ strings with empirical distribution $q$. Each $nq(a)$ must be a nonnegative integer and the counts must sum to $n$; otherwise the class is empty. Distinct type classes partition ${\mathcal{A}}^n$: every string belongs to exactly one class.

For four binary symbols there are five type classes, indexed by the number $k$ of ones. Their cardinalities are $1,4,6,4,1$, which sum to $16=2^4$. The balanced class contains exactly $0011,0101,0110,1001,1010,1100$. These are six different strings, not six different empirical distributions.