Related papers: Another proof of Harer-Zagier formula
By pairwise gluing of edges of a polygon, one produces two-dimensional surfaces with handles and boundaries. In this paper, we count the number ${\cal N}_{g,L}(n_1, n_2, ..., n_L)$ of different ways to produce a surface of given genus $g$…
The goal of this note is to provide a very short proof of Harer-Zagier formula for the number of ways of obtaining a genus g Riemann surface by identifying in pairs the sides of a (2d)-gon, using semi-infinite wedge formalism operators.
The study of bipartite maps (or Grothendieck's dessins d'enfants) is closely connected with geometry, mathematical physics and free probability. Here we study these objects from their permutation factorization formulation using a novel…
Harer and Zagier proved a recursion to enumerate gluings of a $2d$-gon that result in an orientable genus $g$ surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other…
In this paper the number of ways to glue together several polygons into a surface of genus $g$ has been investigated. We've given an elementary proof on the formula for the generating function $\mathbf{C}_g^{[2]}(z)$ of the number of…
In the symmetric group on a set of size 2n, let P_{2n} denote the conjugacy class of involutions with no fixed points (equivalently, we refer to these as ``pairings'', since each disjoint cycle has length 2). Harer and Zagier explicitly…
In this paper the number of ways to glue a surface of genus $g$ has been investigated. We've proven formulas for the number of gluings sphere from three polygons and from two bicolored polygons. Moreover, we've given a new proofs on the…
We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. We also give a weighted variant for the generating polynomial $Q_g^n(x)$ where $x$ is a parameter taking the number of…
Given an identity relating families of Schur and power sum symmetric functions, this may be thought of as encoding representation-theoretic properties according to how the $p$-to-$s$ transition matrices provide the irreducible character…
Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank - which generalize the…
The Hultman numbers enumerate permutations whose cycle graph has a given number of alternating cycles (they are relevant to the Bafna-Pevzner approach to genome comparison and genome rearrangements). We give two new interpretations of the…
We construct two series of commuting Hamiltonians parametrized by a constant matrix. The first series was my guess and the second one was know, and in our approach follows from the first series. For the proof we use familier facts known…
In this paper, we begin with the Lehman-Walsh formula counting one-face maps and construct two involutions on pairs of permutations to obtain a new formula for the number $A(n,g)$ of one-face maps of genus $g$. Our new formula is in the…
On the background of Zhang's local Gross-Zagier formulae for GL(2), we study some p-adic problems. The local Gross-Zagier formulae give identities of very special local geometric data (local linking numbers) with certain local Fourier…
Let $G_n$ be the genus of a two-dimensional surface obtained by gluing, uniformly at random, the sides of an $n$-gon. Recently Linial and Nowik proved, via an enumerational formula due to Harer and Zagier, that the expected value of $G_n$…
Mock modular forms have their origins in Ramanujan's pioneering work on mock theta functions. In a 1975 paper, Zagier proved certain transformation properties of the generating function of the Hurwitz class numbers $H(n)$ for the…
Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$ whose characteristic does not equal 2. We give criteria for the existence of a curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the products of…
By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for $\zeta(2+r+2s)$ (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such…
Exact recursion formulas for mixed moments of four fundamental random matrix ensembles are derived. The reason such recursive formulas are possible is closely related to properties of polygon gluings studied by Harer and Zagier as well as…
We prove three fundamental properties of counting holomorphic cylinders in log Calabi-Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual…