Symmetry of meromorphic differentials produced by involution identity and relation to integer partitions

We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on $$\mathbb{C }\!\mathrm{P }^1$$ and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees.

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

formula omitted]-torsion points in finite abelian groups and combinatorial identities

The n th-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n . Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.

A class of combinatorial identities

Two loop-free algorithms, LEX and NEXPAR2, for generating the partitions of a positive integer n, in lexicographic and antilexicographic order, respectively, are investigated, and their structures are shown to be closely related. By utilising a number of combinatorial identities which relate the cardinalities of various classes of partitions, we derive formulae from which operation counts for the two algorithms may be obtained. For large n, corresponding asymptotic formulae are also obtained. A number of modifications and refinements of the two algorithms are discussed, and the relative performance of the algorithms resulting, and also the original algorithms, is analysed for both the non-asymptotic and the asymptotic cases.

A partition identity

The KOH terms and classes of unimodal N-modular diagrams

One basic activity in combinatorics is to establish combinatorial identities by so-called `bijective proofs,' which amounts to constructing explicit bijections between two types of the combinatorial objects in question. The aim of this paper is to show how techniques from the formal logic world can be applied directly to such problems studied completely independently in the world of combinatorics. The basic idea is to characterize equinumerous partition ideals in terms of the minimal elements generating the order filters, the complements to the original ideals. For the case where the minimal elements of each of these order filters are disjoint, we have developed a general automated method to remove the mystery of certain known results and establish new results in the theory of integer partitions: Kanovich [Finding direct partition bijections by two-directional rewriting techniques. Discrete Math., 285 (1-3) (2004) 151-166], and Kanovich [The two-way rewriting in action: Removing the mystery of Euler-Glaisher's map. Discrete Math., (2006), doi:10.1016/j.disc.2006.10.005]. Here we address the case of filters having overlapping minimal elements. Essentially this is a case study related to the Fibonacci numeration system. In addition to a `bijective proof' for Zeckendorf's theorem - that every positive integer is uniquely representable within the Fibonacci numeration system, we establish `bijective proofs' for a new series of partition identities related to Fibonacci and Lucas numbers. The main result is proved using the idea of filter supports, and rewriting systems on multisets having overlapping patterns. The main technical step is a suitable construction of such a rewriting system and then a proof that this rewriting system and the system consisting of its reverse rewriting rules, both have the Church-Rosser property, with providing the bijections we are looking for.