The Art of Bijective Combinatorics Part IV
Combinatorial theory of orthogonal polynomials and continued fractions
The Institute of Mathematical Sciences, Chennai, India (January-March 2019)
Lectures related to the course
Laguerre heaps of segments for the PASEP
algebra and combinatorics seminar IISc, Bangalore, March 8, 2019
slides (pdf, 29 Mo)
The PASEP (partially asymmetric exclusion process) is a toy model in the physics of dynamical systems strongly related to the moments of some classical orthogonal polynomials (Hermite, Laguerre, Askey-Wilson). The partition function has been interpreted with various combinatorial objects such as permutations, alternative and tree-like tableaux, etc. We introduce a new one called "Laguerre heaps of segments", which seems to play a central role in the network of bijections relating all these interpretations.
A survey of the combinatorial theory of orthogonal polynomials and continued fractions.
colloquium IISc, Bangalore, March 7, 2019
slides (pdf, 23 Mo)
The theory of orthogonal polynomials started with analytic continued fractions going back to Euler, Gauss, Jacobi, Stieljes ... The combinatorial interpretations started in the late 70's and is a an active research domain. I will give the basis of the theory interpreting the moments of general (formal) orthogonal polynomials, Jacobi continued fractions and Hankel determinants with some families of weighted paths. In a second part I will give some examples of interpretations of classical orthogonal polynomials and of their moments (Hermite, Laguerre, Jacobi, ...) with their connection to theoretical physics.
Proofs without words: the example of the Ramanujan continued fraction
colloquium IMSc, Chennai, February 21, 2019
slides (pdf, 28 Mo) video link to YouTube (from IMSc Matsciencechanel Playlist)
Visual proofs of identities were common at the Greek time, such as the Pythagoras theorem. In the same spirit, with the renaissance of combinatorics, visual proofs of much deeper identities become possible. Some identities can be interpreted at the combinatorial level, and the identity is a consequence of the construction a weight preserving bijection between the objects interpreting both sides of the identity.
In this lecture, I will give an example involving the famous and classical Ramanujan continued fraction. The construction is based on the concept of "heaps of pieces",
which gives a spatial interpretation of the commutation monoids introduced by Cartier and Foata in 1969.