JournalsaihpdVol. 6, No. 4pp. 607–640

Enumerating meandric systems with large number of loops

  • Motohisa Fukuda

    Yamagata University, Japan
  • Ion Nechita

    TU München, Germany and Université de Toulouse, France
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Abstract

We investigate meandric systems with a large number of loops using tools inspired by free probability. For any fixed integer r, we express the generating function of meandric systems on 2n2n points with nrn \to r loops in terms of a finite (the size depends on rr) subclass of irreducible meandric systems, via the moment-cumulant formula from free probability theory. We show that the generating function, after an appropriate change of variable, is a rational function, and we bound its degree. Exact expressions for the generating functions are obtained for r6r \leq 6, as well as the asymptotic behavior of the meandric numbers for general rr.

Cite this article

Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of loops. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), no. 4, pp. 607–640

DOI 10.4171/AIHPD/80