Algebraic and Enumerative Combinatorics

Speaker : Yvonne KARIUKI (Kibabii University,Kenya),Fufa BEYENE (Kotebe University of Education,Ethiopia)

This course provides a foundational introduction to enumerative combinatorics. Core topics include basic counting principles (sum and product rules), permutations and combinations, the inclusion–exclusion principle, recurrence relations, generating functions, partitions, and bijective proofs. Selected advanced topics such as Catalan numbers, and Stirling and Bell numbers may also be introduced. Time split: 2 hours interactive lectures, 1 hour exercises, 2 hours interactive lectures, 1 hour exercises. Note that Yvonne Kariuki and Fufa Beyene will share teaching this course (with both active in the lectures - engaging with students and teaching), and so the session total duration is twice the teaching time.

Speaker : Clement REQUILE (Universitat Politècnica de Catalunya (UPC - BarcelonaTech),Spain)

This course will serve as an introduction to determinental methods in combinatorics, which play an important rôle in enumerative combinatorics, in particular in graph theory and its connections with physics. After recalling some properties of the determinant in the context of graph theory, in particular how to count walks in graphs, we will start with the famous matrix-tree theorem of Kirchoff, relating the number of spanning trees with a determinant formula, and some of its generalisations due notably to Tutte and motivated by the conductance of electrical networks. We will then introduce the Lindström-Gessel-Viennot lemma, which generalises the matrix-tree theorem in the sense that it allows us to count walks in directed graphs using a determinant. And we will briefly discuss the deep connections of this method with some statistical physics models. Finally, we will present Kasteleyn's theorem and how a determinant formula due to Cayley allows us to compute the number of perfect matchings in planar graphs, and in what way this question is related to the dimer and the Ising models in statistical physics. Time split: 2 hours interactive lectures, 1 hour exercises, 2 hours interactive lectures, 1 hour exercises.

Speaker : Dimbinaina RALAIVAOSAONA (Stellenbosch University,South Africa)

The course aims to introduce classical algebraic enumeration methods, with examples of combinatorial structures that are relevant in disciplines such as computer science and biology. Topics include bijections, symbolic method, various types of generating functions, Lagrange inversion, and Pólya’s enumeration method. Students will learn to apply these methods to enumerate combinatorial structures such as permutations, trees, tanglegrams, directed graphs, and more. Time split: 2 hours interactive lectures, 1 hour exercises, 2 hours interactive lectures, 1 hour exercises.

Speaker : Alex Samuel BAMUNOBA (Lira University,Uganda)

In this course we will discuss polynomials, recursions, and q-analogs: connecting the enumerative combinatorics background in the first week to algebraic concepts that will be discussed during the second week. Polynomials can be used to capture a number of algebraic graph invariants, we will focus on the chromatic polynomial and the Tutte polynomial. Recursive relationships can be used to describe these polynomials and these will be explored. Then, we will discuss q-analogs and how they can be used to perform refined enumeration. Time split: 2 hours interactive lectures, 1 hour exercises, 2 hours interactive lectures, 1 hour exercises.

Speaker : Nancy NEUDAUER (Pacific University,United States)

Matroids show up several times in the undergraduate curriculum, but most of us don’t know them by name. In 1933, three Harvard junior-fellows tied together some recurring themes in mathematics, into what Gian Carlo Rota called one of the most important ideas of our day. They were finding properties of dependence in multiple mathematical structures. What resulted is the matroid, which abstracts notions of algebraic dependence, linear independence, and geometric dependence, thus unifying several areas of mathematics. The usefulness of matroids to pure mathematical research is similar to that of groups – by studying an abstract version of phenomena that occur in different realms of mathematics, we learn something about all those realms simultaneously. We find that matroids are everywhere: Vector spaces are matroids; We can define matroids on a graph. Matroids are useful in situations that are modelled by both graphs and matrices. Yet many matroids cannot be represented by a graph nor a collection of vectors over any field. We consider the essential role of matroids in combinatorial optimization. No prior knowledge of matroids or graphs is needed. Time split: 2 hours interactive lectures, 1 hour exercises, 2 hours interactive lectures, 1 hour exercises.

Speaker : Stephan WAGNER (Graz University of Technology,Austria)

When exact counting formulas are not available, one often uses analytic tools to obtain approximate and asymptotic solutions to various combinatorial problems. This course provides an introduction to asymptotic analysis and its use in the context of combinatorics. Important techniques from real analysis (e.g., Laplace's method or the Euler-Maclaurin formula) and complex analysis (e.g., singularity analysis and the saddle point method) will be presented and illustrated with topical examples from enumerative combinatorics. Time split: 2 hours interactive lectures, 1 hour exercises, 2 hours interactive lectures, 1 hour exercises.

We introduce the space of symmetric functions and the most common bases for this space. We prove several formulas formulas for the Schur functions as well as give combinatorial interpretations for some of the transition matrices between the different bases (Kostka coefficients and the Murnaghan--Nakayama rule). This part also builds on the Lindström-Gessel-Viennot lemma (introduced earlier in the school). Given time, we also discuss the famous RSK-algorithm and the Littlewood-Richardson coefficients, methods for proving Schur positivity, and connection with quasisymmetric functions. Time split: 2 hours interactive lectures, 1 hour exercises, 2 hours interactive lectures, 1 hour exercises.

Speaker : Viviane PONS (Paris-Saclay University,France)

We explore the definition of partial orders and lattices based on some famous examples from combinatorics such as the weak order on permutations and the Tamari lattice. We explore specific properties (distributivity, demi-distributivity, congruence uniform) and operations (sub lattices and lattice quotient). Time split: 2 hours interactive lectures, 1 hour exercises, 2 hours interactive lectures, 1 hour exercises.