Related papers: Rationality in map and hypermap enumeration by gen…
This contribution summarizes recent work of the authors that combines methods from dynamical systems theory (discrete Painlev\'e equations) and asymptotic analysis of orthogonal polynomial recurrences, to address long-standing questions in…
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…
We define a ring whose elements are rational functions, whose addition is polynomial multiplication, and whose multiplication is a convolution operation. It is then show that this ring's endomorphisms exhibit a strong classification.…
This paper addresses the enumeration of rooted and unrooted hypermaps of a given genus. For rooted hypermaps the enumeration method consists of considering the more general family of multirooted hypermaps, in which darts other than the root…
We produce an infinite family of transcendental numbers which, when raised to their own power, become rational. We extend the method, to investigate positive rational solutions to the equation $x^x = \alpha$, where $\alpha$ is a fixed…
In this paper, we give the first combinatorial proof of a rationality scheme for the generating series of maps in positive genus enumerated by both vertices and faces, which was first obtained by Bender, Canfield and Richmond in 1993 by…
It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
Consider a subfield of the field of rational functions in several indeterminates. We present an algorithm that, given a set of generators of such a subfield, finds a simple generating set. We provide an implementation of the algorithm and…
We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well quasi-ordered has a rational generating function. To do so we show that any such class is…
For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After…
A recursive method is given for finding generating functions which enumerate rooted hypermaps by number of vertices, edges and faces for any given number of darts. It makes use of matrix-integral expressions arising from the study of…
If the denominator of a rational function of several variables is sum of even powers and the numerator is a monomial, then we give a numerical criterion, using the exponents involved in the expression of the rational function, to decide if…
We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input…
We develop a nonstandard approach to exploring polynomials associated with peaks and runs of permutations. With the aid of a context-free grammar, or a set of substitution rules, one can perform a symbolic calculus, and the computation…
One develops {\em ab initio} the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A numerical invariant of a rational map is introduced, called the Jacobian…
Rationale is defined as a subset of input features that best explains or supports the prediction by machine learning models. Rationale identification has improved the generalizability and interpretability of neural networks on vision and…
Given a holonomic sequence $F(n)$, we characterize rational functions $r(n)$ so that $r(n)F(n)$ can be summable. We provide upper and lower bounds on the degree of the numerator of $r(k)$ and show the denominator of $r(n)$ can be read from…
One defines two ways of constructing rational maps derived from other rational maps, in a characteristic-free context. The first introduces the Newton complementary dual of a rational map. One main result is that this dual preserves…
In standard construction of hyperrational numbers using an ultrapower we assume that the ultrafilter is selective. It makes possible to assign real value to any finite hyperrational number. So, we can consider hyperrational numbers with…