相关论文: An algorithm for map enumeration

We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…

Using connections to random matrix theory and orthogonal polynomials, we develop a framework for obtaining explicit closed-form formulae for the number, $\mathscr{N}_{g}(2\nu,j)$, of connected $2\nu$-valent labeled graphs with $j$ vertices…

The aim of the paper is to clarify the nature of combinatorial structures associated with maps on closed compact surfaces. We prove that maps give rise to Lagrangian matroids representable in a setting provided by cohomology of the surface…

We give an expansion in $1/N$ and $\beta$ of the cumulants of power sums of the particles of the $\beta$-ensemble. This new expansion is obtained using the tridiagonal model of Dumitriu and Edelman. The coefficients of the expansion are…

In this note, we examine how the BKP structure of the generating series of several models of maps on non-oriented surfaces can be used to obtain explicit and/or efficient recurrence formulas for their enumeration according to the genus and…

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice…

We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $\rho(g-1)$ for some integer $\rho\ge 1$, and determine isogeny decompositions of the corresponding Jacobian…

By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$.…

A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of fixed genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the…

We enumerate rooted 2-connected and 3-connected surface maps with respect to vertices and edges. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive…

This summarizes our latest understanding and results about the algorithms for enumerating Tanner Graphs that have a regular structure called Balanced Tanner Graphs. Enumeration algorithms for Balanced Tanner Graphs based upon Cyclic…

The enumeration of maps and the study of uniform random maps have been classical topics of combinatorics and statistical physics ever since the seminal work of Tutte in the sixties. Following the bijective approach initiated by Cori and…

Ihe first author presented an efficient algorithm for computing involutive (and reduced Groebner) bases. In this paper, we consider a modification of this algorithm which simplifies matters to understand it and to implement. We prove…

Hypermaps were introduced as an algebraic tool for the representation of embeddings of graphs on an orientable surface. Recently a bijection was given between hypermaps and indecomposable permutations; this sheds new light on the subject by…

A unicellular map, or one-face map, is a graph embedded in an orientable surface such that its complement is a topological disk. In this paper, we give a new viewpoint to the structure of these objects, by describing a decomposition of any…

We present a fast enumeration algorithm for combinatorial 2- and 3-manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3-manifolds with 11 vertices. We further determine all…

We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be…

Restricted Boltzmann Machines (RBMs) and models derived from them have been successfully used as basic building blocks in deep artificial neural networks for automatic features extraction, unsupervised weights initialization, but also as…

We present an algorithm for computing equations of canonically embedded Riemann surfaces with automorphisms. A variant of this algorithm with many heuristic improvements is used to produce equations of Riemann surfaces $X$ with large…

Let $G$ denote a countable inverse semigroup. We construct a kind of a Baum--Connes map $K(\tilde A \rtimes G) \rightarrow K(A \rtimes G)$ by a categorial approach via localization of triangulated categories, developed by R. Meyer and R.…