I. P. Goulden

The Number of Ramified Coverings of the Sphere by the Torus and Surfaces of Higher Genera

Abstract. We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification points. In addition, the conjecture is proved for simple coverings of the sphere by the torus. We obtain corresponding expressions for surfaces of higher genera for small number of ramification points, and conjecture the general form for this number in terms of a symmetric polynomial that appears to be new. The approach involves the analysis of the action of a transposition to derive a system of linear partial differential equations that give the generating series for the desired numbers.

The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group

A combinatorial bijection is given between pairs of permutations in S n the product of which is a given n -cycle and two-coloured plane edge-rooted trees on n edges, when the numbers of cycles in the disjoint cycle representations of the permutations sum to n + 1. Thus the corresponding connection coefficient for the symmetric group is determined by enumerating these trees with respect to appropriate characteristics. This is extended to the case of m -tuples of permutations in S n the product of which is a given n -cycle, in which the combinatorial objects replacing trees are cacti of m -gons.

A differential operator for symmetric functions and the combinatorics of multiplying transpositions

By means of irreducible characters for the symmetric group, formulas have previously been given for the number of ways of writing permutations in a given conjugacy class as products of transpositions. These formulas are alternating sums of binomial coefficients and powers of integers. Combinatorial proofs are obtained in this paper by analyzing the action of a partial differential operator for symmetric functions.

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 determine various linear recurrence equations for the numbers of these coverings with no ramification over infinity; one of these recurrence equations has previously been conjectured by Graber and Pandharipande. The general form of this series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.

A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves

We determine an expression 9 (1X) for the virtual Euler characteristics of the moduli spaces of s-pointed real (-y = 1/2) and complex (-y = 1) algebraic curves. In particular, for the space of real curves of genus g with a fixed point free involution, we find that the Euler characteristic is (-2)s-1(1-29-1)(g?s-2)!Bg/g! where Bg is the gth Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is (-1) S (g+s-2)!Bg+ /(g+1) (g -1) ! The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to counit cells. The approach involves a parameter y that permits specialization of the formula to the real and complex cases. This suggests that 4 (-y) itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.

Tree-like Properties of Cycle Factorizations

We provide a bijection between the set of factorizations, that is, ordered (n?1)-tuples of transpositions in Sn whose product is (12?n), and labelled trees on n vertices. We prove a refinement of a theorem of J. Denes (1959, Publ. Math. Inst. Hungar. Acad. Sci.4, 63?71) that establishes new tree-like properties of factorizations. In particular, we show that a certain class of transpositions of a factorization corresponds naturally under our bijection to leaf edges (incident with a vertex of degree 1) of a tree. Moreover, we give a generalization of this fact.

Planar decompositions of tableaux and Schur function determinants

Abstract In this paper we describe planar decompositions of skew shape tableaux into strips and use the shapes of these strips to generate a determinant. We then prove that each of these determinants is equal to the Schur function for the skew shape. The Jacobi-Trudi identity, the dual Jacobi-Trudi identity, the Giambelli identity and the rim ribbon identity of Lascoux and Pragacz are all special cases of this theorem. A compact Gessel-Viennot lattice path argument provides the proof.

Graph factorization, general triple systems, and cyclic triple systems

In this self-contained exposition, results are developed concerning one-factorizations of complete graphs, and incidence matrices are used to turn these factorization results into embedding theorems on Steiner triple systems. The result is a constructive graphical proof that a Steiner triple system exists for any order congruent to 1 or 3 modulo 6. A pairing construction is then introduced to show that one can also obtain triple systems which are cyclically generated.

Maintaining the spirit of the reflection principle when the boundary has arbitrary integer slope

We provide a direct geometric bijection for the number of lattice paths that never go below the line y = kx for a positive integer k. This solution to the Generalized Ballot Problem is in the spirit of the reflection principle for the Ballot Problem (the case k = 1), but it uses rotation instead of reflection. It also gives bijective proofs of the refinements of the Generalized Ballot Problem which consider a fixed number of right-up or up-right corners.

Raney Paths and a Combinatorial Relationship between Rooted Nonseparable Planar Maps and Two-Stack-Sortable Permutations

An encoding of the set of two-stack-sortable permutations (TSS) in terms of lattice paths and ordered lists of strings is obtained. These lattice paths are called Raney paths. The encoding yields combinatorial decompositions for two complementary subsets of TSS, which are the analogues of previously known decompositions for the set of nonseparable rooted planar maps (NS). This provides a combinatorial relationship between TSS and NS, and, hence, a bijection is determined between these sets that is different, simpler, and more refined than the previously known bijection.