The number of ramified coverings of the sphere by the double torus, and a general form for higher genera

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a conjecture of Graber and Pandharipande, giving a linear recurrence equation for the number of these coverings with no ramification over infinity. The general form of the series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.Comment: 14pp.; revised version has two additional results in Section

Hurwitz numbers and intersections on moduli spaces of curves

This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.Comment: 30 pages (AMSTeX). Minor typos are correcte

The number of ramified covering of a Riemann surface by Riemann surface

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point.Comment: LaTex, 14 page

Polynomiality of monotone Hurwitz numbers in higher genera

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e., up to a specified combinatorial factor, the monotone Hurwitz number in genus g with ramification specified by a given partition is a polynomial indexed by g in the parts of the partition.Comment: 23 page

Towards studying the structure of triple Hurwitz numbers

Going beyond the studies of single and double Hurwitz numbers, we report some progress towards studying Hurwitz numbers which correspond to ramified coverings of the Riemann sphere involving three nonsimple branch points. We first prove a recursion which implies a fundamental identity of Frobenius enumerating factorizations of a permutation in group algebra theory. We next apply the recursion to study Hurwitz numbers involving three nonsimple branch points (besides simple ones), two of them having complete ramification profiles while the remaining one having a prescribed number of preimages. The recursion allows us to obtain recurrences as well as explicit formulas for these numbers. The case where one of the nonsimple branch points with complete profile has a unique preimage (one-part quasi-triple Hurwitz numbers) is particularly studied in detail. We prove a dimension-reduction formula from which any one-part quasi-triple Hurwitz number can be reduced to quasi-triple Hurwitz numbers where two branch points respectively have a unique preimage. We also obtain the polynomiality of one-part quasi-triple Hurwitz numbers analogous to that implied by the remarkable ELSV formula, suggesting a potential connection to Hodge integrals or intersection theory. Coefficients of the polynomials are completely and explicitly determined which may facilitate searching these integral counterparts (i.e., ELSV-type formulas) in the future.Comment: An extended abstract, comments are very much welcome. The full version with more results and details will be uploaded late

Transitive factorizations of permutations and geometry

We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of curves. Aspects of these seemingly unrelated areas are seen to be related in a unifying view from the perspective of algebraic combinatorics. At several points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th birthda