The boundary length and point spectrum enumeration of partial chord   diagrams using cut and join recursion
  We introduce the boundary length and point spectrum, as a joint generalization of the boundary length spectrum and boundary point spectrum in arXiv:1307.0967. We establish by cut-and-join methods that the number of partial chord diagrams filtered by the boundary length and point spectrum satisfies a recursion relation, which combined with an initial condition determines these numbers uniquely. This recursion relation is equivalent to a second order, non-linear, algebraic partial differential equation for the generating function of the numbers of partial chord diagrams filtered by the boundary length and point spectrum. 
J{\o}rgen Ellegaard Andersen, Hiroyuki Fuji, Robert C. Penner,   Christian M. Reidys
math-ph hep-th math.CO math.MP q-bio.QM