A Formal Calculus for the Enumerative System of Sequences—III. Further Developments

We use the material of Parts I and II to obtain further results in sequence enumeration. These include the enumeration of sequences with respect to both maximal π₁ and π₂-paths, and the enumeration of sequences with distinguished substrings. We use the material to derive an enumerative proof of a generalization of a result of Foata and Schützenberger. Finally, we reconsider the enumeration of permutations with periodic pattern and show that the required generating function may be exhibited as the solution of a set of first-order nonlinear differential equations. Subsequent work has shown these to be matrix Riccati equations, although we refer to them here as Volterra equations.     

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.     

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

We determine an expression for the virtual Euler characteristics of the moduli spaces of -pointed real ) and complex () algebraic curves. In particular, for the space of real curves of genus with a fixed point free involution, we find that the Euler characteristic is where is the th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is

The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter that permits specialization of the formula to the real and complex cases. This suggests that itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.

     

The Moduli Space of Curves, Double Hurwitz Numbers, and Faber��s Intersection Number Conjecture

We define the dimension 2g − 1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of \mathbbP₁{{\mathbb{P}_1}} with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by “top intersections” of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a “combinatorialization” of top intersections of Y{\Psi} -classes. As genus 0 double Hurwitz numbers with at most 3 parts over ∞ are well understood, we obtain Faber’s Intersection Number Conjecture for up to 3 parts, and an approach to the Conjecture in general (bypassing the Virasoro Conjecture). We also recover other geometric results in a unified manner, including Looijenga’s theorem, the socle theorem for curves with rational tails, and the hyperelliptic locus in terms of κ _g–2.     

Enumerative Properties of NC (B) (p,q)

We determine the rank generating function, the zeta polynomial and the M?bius function for the poset NC ^(B) (p, q) of annular non-crossing partitions of type B, where p and q are two positive integers. We give an alternative treatment of some of these results in the case q = 1, for which this poset is a lattice. We also consider the general case of multiannular noncrossing partitions of type B, and prove that this reduces to the cases of non-crossing partitions of type B in the annulus and the disc.     

Natural exact covering systems and the reversion of the Möbius series

The Ramanujan Journal - We prove that the number of natural exact covering systems of cardinality k is equal to the coefficient of $$x^k$$ in the reversion of the power series $$\sum _{k \ge 1} \mu...     

Transitive Factorizations in the Symmetric Group, and Combinatorial Aspects of Singularity Theory

We consider the determination of the number c_k(α) of ordered factorizations of an arbitrary permutation on n symbols, with cycle distribution α, intok -cycles such that the factorizations have minimal length and the group generated by the factors acts transitively on then symbols. The case k =  2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the full symmetric group. For k =  3, the monodromy group is the alternating group, and this is another case that, in principle, is of considerable interest. We conjecture an explicit form, for arbitrary k, for the generating series for c_k(α), and prove that it holds for factorizations of permutations with one, two and three cycles (so α is a partition with at most three parts). Our approach is to determine a differential equation for the generating series from a combinatorial analysis of the creation and annihilation of cycles in products under the minimality condition.     

Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions

A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in , the Jack parameter shifted by . More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families.

The coefficients of a second series, essentially the logarithm of the first, specialize at values and of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in , and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.

     

Mass spectra of esters of α-hydroxy and α-methoxy acids

The most abundant fragment produced by electron bombardment of esters of the type R¹R²C(OR³)CO₂R⁴ is the R¹R²C = \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\rm O}\limits^{{\rm + } \cdot } $\end{document}R³ ion. Methyl glycollate (R¹ = R² = R³ = H, R⁴ = Me) eliminates the HCO˙ radical by a complex rearrangement involving the methylenic hydrogen atoms. The methyl and ethyl esters of methoxyacetic acid (R¹ = R² = H, R³ = Me, R⁴ = Me or Et) eliminate formaldehyde by the McLafferty rearrangement.