Related papers: Philippe Flajolet's early work in combinatorics
Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has…
Let q be a power of a prime and E be an elliptic curve defined over F_q. In "Combinatorial aspects of elliptic curves" [17], the present author examined a sequence of polynomials which express the N_k's, the number of points on E over the…
We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix…
Contribution of Indian Mathematics since Vedic Period has been recognised by the Historians. Pingala (200 BC) in his book on 'Chandashaastra', a text related to the description and analysis of meters in poetic work, describes algorithms…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
We cover some useful techniques in computational aspects of analytic number theory, with specific emphasis on ideas relevant to the evaluation of L-functions. These techniques overlap considerably with basic methods from analytic number…
This set of notes is based on a lecture I gave at "50 years of Finite Geometry | A conference on the occasion of Jef Thas's 70th birthday," in November 2014. It consists essentially of three parts: in a first part, I introduce some ideas…
Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and…
This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings,…
We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Fa\`a di Bruno formula,…
The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in \emph{geometric invariant theory}. The concept of \emph{observable subgroup} was introduced in the early 1960s…
We prove that the existence of finite combinatorial objects such as affine planes, mutually orthogonal Latin squares, and resolvable balanced incomplete block designs can be reformulated as the existence of certain algorithmic reductions…
In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…
We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we…
In this paper we study the problem of estimating the alpha-, beta- and phi-mixing coefficients between two random variables, that can either assume values in a finite set or the set of real numbers. In either case, explicit closed-form…
In this paper, we explore applications of combinatorics on words across various domains, including data compression, error detection, cryptographic protocols, and pseudorandom number generation. The examination of the theoretical…
In this paper we reformulate in a simpler way the combinatoric core of constructive quantum field theory We define universal rational combinatoric weights for pairs made of a graph and one of its spanning trees. These weights are nothing…
Primary definitions, notation and general observations of finite fibonomial operator calculus (ffoc) are presented. Kwasniewski's combinatorial interpretation of fibonomial coefficients by the use of fibonacci cobweb poset is given. Some…
Nature rarely reveals her secrets bluntly, yet in the Fibonacci sequence she grants us a glimpse of her quiet architecture of growth, harmony, and recursive stability \citep{Koshy2001Fibonacci, Livio2002GoldenRatio}. From spiral galaxies to…