  Using combinatorial techniques, we answer two questions about simple
classical Lie groups. Define $N(G,m)$ to be the number of conjugacy classes of
elements of finite order $m$ in a Lie group $G$, and $N(G,m,s)$ to be the
number of such classes whose elements have $s$ distinct eigenvalues or
conjugate pairs of eigenvalues. What is $N(G,m)$ for $G$ a unitary, orthogonal,
or symplectic group? What is $N(G,m,s)$ for these groups? For some cases, the
first question was answered a few decades ago via group-theoretic techniques.
It appears that the second question has not been asked before; here it is
inspired by questions related to enumeration of vacua in string theory. Our
combinatorial methods allow us to answer both questions.
