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We show that plane bipolar posets (i.e., plane bipolar orientations with notransitive edge) and transversal structures can be set in correspondence tocertain (weighted) models of quadrant walks, via suitable specializations of abijection due to Kenyon, Miller, Sheffield and Wilson. We then derive exact andasymptotic counting results. In particular we prove (computationally and thenbijectively) that the number of plane bipolar posets on $n+2$ vertices equalsthe number of plane permutations of size $n$. Regarding transversal structures,for each $v\geq 0$ we consider $t_n(v)$ the number of such structures with$n+4$ vertices and weight $v$ per quadrangular inner face (the case $v=0$corresponds to having only triangular inner faces). We obtain a recurrence tocompute $t_n(v)$, and an asymptotic formula that for $v=0$ gives $t_n(0)\sim c\\!(27/2)^nn^{-1-\pi/\mathrm{arccos}(7/8)}$ for some $c>0$, which also ensuresthat the associated generating function is not D-finite.

On the enumeration of plane bipolar posets and transversal structures