Combinatorics of the tautological lamination

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 m q -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 ⌊n/2⌋ < m < n.