Enumeration of rooted constellations and hypermaps through quantum matrix integrals

• Main Author: Dyer, Jacob
• Other Authors: Kay, Bernard
• Published: University of York 2015
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.678781

We present a new method for enumeration of rooted constellations and other objects these can represent, specifically rooted hypermaps and maps. We derive a closed-form generating function enumerating rooted hypermaps with one face and a fixed number of darts, partitioned by number of edges, and vertices. We derive an algorithmic procedure for calculating generating functions enumerating all rooted hypermaps for fixed number of darts, partitioning by number of edges, vertices and faces, as well as an analogous procedure for enumerating rooted maps for fixed edge count. We also look at the enumeration problem for general rooted constellations, but do not calculate generating functions. Using these results we find recursion relations for calculating the total number of rooted hypermaps, maps and constellations of any given degree. This method is based on matrix integration tools originally developed in the study of bipartite quantum systems, specifically in calculating mean properties of their subsystems, where the averaging is over all possible pure states of the overall system. We present this work first, studying the mean von Neumann entropy of entanglement between the quantum system's two subsystems. We look at an unproven entropy approximation proposed by Lubkin (1978), derived from an infinite series expansion of the entropy which was not known to be convergent. We prove that this series is convergent if and only if the subsystem being studied is of dimension two, by deriving closed-form expressions for the series terms and finding their limiting behaviour. In light of this we examine the validity of Lubkin's approximation rigorously, confirming the limit in which it is valid, but deriving a more accurate approximation in the process.