A lamination L is a closed set of chords of the unit disk so that no two chords intersect in the open disk. A lamination is d-invariant under the degree d covering map $\sigma_d:S\to S$ of the unit circle if it is forward invariant (for any chord $ab$ in L the chord $\sigma(a)\sigma(b)$ is also in $L$). In this talk we will discuss properties of d-invariant laminations that all contain a given forward invariant subset $P$ of chords (for example a given periodic chord).

We count possible preimage laminations for n steps. i.e. the number of laminations that have a particular $P$ as their $\sigma_d^n$ image. In contrast to most laminations research, we do not specify critical cords (i.e., chords $ab$ so that $\sigma(a)=\sigma(b)$).

We define what laminations should be included in our count. Particularly, we exclude critical and degenerate leaves from our laminations because they make the count immediately infinite. We also insist that each of the counted laminations are maximal, to avoid confluence, and have adequately many chords with the same image.

This class of laminations has the added advantage that they are all realized by complex polynomials of degree d, giving us some hope that we can use our combinatorial model to assemble a model of polynomial parameter space. It is clear in the degree 2 case that the laminations which we generate in our count correspond to limbs outside the molecule of the connectedness locus, with exactly one exception for each n.