On the enumeration of plane bipolar posets and transversal structures

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 (computationally and then bijectively) that the number of plane bipolar posets on n + 2 vertices equals the number of plane permutations (i.e., avoiding the vincular pattern 2 14 3) of size n, and that the number tn of transversal structures on n + 2 vertices satisfies (for some c > 0) the asymptotic estimate tn ∼ c (27/2)nn−1−π/arccos(7/8), which also ensures that the associated generating function is not D-finite.

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