Hall-Littlewood polynomials and Cohen-Lenstra heuristics for Jacobians of random graphs

Cohen-Lenstra heuristics for Jacobians of random graphs give rise to random partitions. We connect these random partitions to the Hall-Littlewood polynomials of symmetric function theory, and use this connection to give combinatorial proofs of properties of these random partitions. In addition, we use Markov chains to give an algorithm for generating these partitions.Comment: 10 page

Belief propagation in monoidal categories

We discuss a categorical version of the celebrated belief propagation algorithm. This provides a way to prove that some algorithms which are known or suspected to be analogous, are actually identical when formulated generically. It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810

Geometric generalizations of the Tonnetz and their relation to Fourier phase space

Some recent work on generalized Tonnetze has examined the topologies resulting from Richard Cohn’s common-tone based formulation, while Tymoczko has reformulated the Tonnetz as a network of voice-leading relationships and investigated the resulting geometries. This paper adopts the original common-tone based formulation and takes a geometrical approach, showing that Tonnetze can always be realized in toroidal spaces,and that the resulting spaces always correspond to one of the possible Fourier phase spaces. We can therefore use the DFT to optimize the given Tonnetz to the space (or vice-versa). I interpret two-dimensional Tonnetze as triangulations of the 2-torus into regions associated with the representatives of a single trichord type. The natural generalization to three dimensions is therefore a triangulation of the 3-torus. This means that a three-dimensional Tonnetze is, in the general case, a network of three tetrachord-types related by shared trichordal subsets. Other Tonnetze that have been proposed with bounded or otherwise non-toroidal topologies, including Tymoczko’s voice-leading Tonnetze, can be under-stood as the embedding of the toroidal Tonnetze in other spaces, or as foldings of toroidal Tonnetze with duplicated interval types.Accepted manuscrip