On the enumeration of plane bipolar posets and transversal structures

Eric Fusy (1) , Erkan Narmanli (2) , Gilles Schaeffer (2)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

We show that plane bipolar posets (i.e., plane bipolar orientations with no transitive edge) and transversal structures can be set in correspondence to certain (weighted) models of quadrant walks, via suitable specializations of a bijection due to Kenyon, Miller, Sheffield and Wilson. We then derive exact and asymptotic counting results, and in particular we prove that the number tn of transversal structures on n + 2 vertices satisfies (for some c > 0) tn ∼ c (27/2) n n −1−π/arccos(7/8) , which also ensures that the associated generating function is not D-finite.

Mots clés: bijections; oriented planar maps; quadrant walks

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Eric Fusy, Erkan Narmanli, Gilles Schaeffer. On the enumeration of plane bipolar posets and transversal structures. European Conference on Combinatorics, Graph Theory and Applications, Sep 2021, Barcelona (on line), Spain. ⟨10.1007/978-3-030-83823-2_90⟩. ⟨hal-03388045⟩