## 4th Algorithmic and Enumerative CombinatoricsSummer School 2018

### Invited Speakers

• François Bergeron (Université du Québec à Montréal, Canada)

Combinatorial enumeration using symmetric functions, with computer algebra exploration
Abstract: Symmetric functions provide one of the most efficient tools for combinatorial enumeration, in the context of objects that may be acted upon by permutations. Only assuming a basic knowledge of linear algebra, we will introduce and describe the toolbox of symmetric functions; and give many interesting examples of their uses. In particular, we will see that many classical formulas of enumerative combinatorics afford a natural generalization in terms of symmetric functions. This will come together with several proposed experimental explorations using computer algebra tools (using SageMath; or Maple, Mathematica, etc.). We will also mention some accessible open (but hard) problems, including some related to algebraic "versions" of P vs NP; as well as a few open problems that may be more manageable.

Further information (including references and software preparation) can be found here .

• Éric Fusy (Laboratoire d'informatique de l'École polytechnique, France)

Planar maps: bijections and applications
Abstract: I will introduce planar graphs and planar maps (which are embedded planar graphs). Planar maps enjoy very nice enumerative properties, as discovered by Tutte in the 1960's, and these results can now be explained thanks to bijective constructions. I will present some of these bijections and the associated tools (in particular certain orientations on planar maps), and will show a few applications, in particular for the geometric representation of planar graphs and the study of distance parameters in random planar maps.

• George Labahn (University of Waterloo, Canada)

Order Bases : Applications and Computation
Abstract: Order Bases take as input a vector or matrix of power series F and describes all solutions (as a module) for approximation problems of the form F p = O(zω) with ω a scalar or a vector. These approximation problems date back to the work of Hermite and his student Padé and later contributions for Order bases were given by Mahler. More recently applications of Order bases to problems in Combinatorics have appeared through the work of Salvy and Bostan. In these lectures we give the history (basically coming from rational approximation and interpolation problems), fast algorithms for computation and applications. The applications will include fast computation of problems with matrix polynomial arithmetic, matrix normal forms in addition to the problems arising in Combinatorics.