= Type class
{title2=$T_n(q)$}
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>.
\Image[/type-classes-four-binary-symbols.png]
{title=Sixteen binary strings form five type classes}
{description=An exact enumeration of all four-symbol strings. Each column fixes the number of ones; the central column has six orderings of the same <empirical distribution>.}
Back to article page
