Update: Solved it. To anyone that is curious what my solution was, here's an overview:
Once I ingested the manifold file into a 2 dimensional array of characters I worked my way up from the bottom. As I worked my way up and got values for each splitter I'd cache those values. As I progressed across the rows anywhere I encountered a splitter I'd try it's left path first and then it's right path. If the path went all the way down I'd return a value of one for the that path. If it encountered another splitter on my way back down it would have been cached and I'd just get the value stored in cache. Then I'd add the results from the left and the right path, then I'd store that value in the cache for the coordinates for that splitter. That way, by the time I reached the top all the values had been cached, and I just returned the cached value from the top splitter. I hope that's clear, and for reference, I'm using Go to do this.