Connected chord diagrams and bridgeless maps
Abstract
We present a surprisingly new connection between two wellstudied combinatorial 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 to Arquès and Béraud for the generating function counting rooted maps with respect to edges and vertices, giving a new bijective 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2016
 DOI:
 10.48550/arXiv.1611.04611
 arXiv:
 arXiv:1611.04611
 Bibcode:
 2016arXiv161104611C
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 63 pages