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Connected chord diagrams and bridgeless maps

Julien Courtiel 1 Karen Yeats 2 Noam Zeilberger 3
1 Equipe AMACC - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
3 PARSIFAL - Proof search and reasoning with logic specifications
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : We present a surprisingly new connection between two well-studied combinato-rial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two classes, which naturally extends to indecomposable diagrams and general rooted maps. As an application, this bijection provides a simplifying framework for some technical quantum field theory work realized by some of the authors. Most notably, an important but technical parameter naturally translates to vertices at the level of maps. We also give a combinatorial proof to a formula which previously resulted from a technical recurrence, and with similar ideas we prove a conjecture of Hihn. Independently, we revisit an equation due toArqù es and Béraud for the generating function counting rooted maps with respect to edges and vertices, giving a new bi-jective interpretation of this equation directly on indecomposable chord diagrams, which moreover can be specialized to connected diagrams and refined to incorporate the number of crossings. Finally, we explain how these results have a simple application to the combinatorics of lambda calculus, verifying the conjecture that a certain natural family of lambda terms is equinumerous with bridgeless maps.
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Submitted on : Tuesday, November 12, 2019 - 5:12:12 PM
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Julien Courtiel, Karen Yeats, Noam Zeilberger. Connected chord diagrams and bridgeless maps. The Electronic Journal of Combinatorics, Open Journal Systems, 2019, 26 (4), pp.1-56. ⟨10.37236/7400⟩. ⟨hal-01650141v2⟩



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