Ion Nechita (Univ. Toulouse and CNRS): Enumerating meanders - three perspectives

Ion Nechita (Univ. Toulouse and CNRS): Talk 2, Session 1, Day 2 of the conference Operator Algebras in Istanbul, 2021, June 9, 2021.

The problem of enumerating meanders is a long-standing open problem in combinatorics. Many different techniques have been used to provide bounds on the asymptotic growth rate of the number of meanders. Here, we present some of the old methods and some new ones, coming from three (related) points of view. First, as noted by Fukuda and Sniady, meanders appear in relation to the partial transposition operation in quantum information theory. A second model for meandric numbers comes from random matrix theory: we shall review some old models due to di Francesco and present some new ones. Finally, I shall present a joint work with Motohisa Fukuda (arXiv:1609.02756 and arXiv:2103.03615) on a third point of view, that of non-commutative probability. Using the operations of free and boolean moment-cumulant transforms, we enumerate large sub-classes of meanders, generalizing previous work of Goulden, Nica, and Puder.