  The Tautological Lamination arises in holomorphic dynamics as a combinatorial
model for the geometry of 1-dimensional slices of the Shift Locus. In each
degree $q$ the tautological lamination defines an iterated sequence of
partitions of $1$ (one for each integer $n$) into numbers of the form $2^m
q^{-n}$. Denote by $N_q(n,m)$ the number of times $2^mq^{-n}$ arises in the
$n$th partition. We prove a recursion formula for $N_q(n,0)$, and a gap
theorem: $N_q(n,n)=1$ and $N_q(n,m)=0$ for $\lfloor n/2 \rfloor < m < n$.

Dynamical Systems;Combinatorics