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 of size $n$, and that the number $t_n$ of transversal structures on $n+2$ vertices satisfies (for some $c>0$) the asymptotic estimate $t_n\sim c\ \!(27/2)^nn^{-1-\pi/\mathrm{arccos}(7/8)}$, which also ensures that the associated generating function is not D-finite.
翻译:我们展示了平面双极形( 平面双极形, 没有中转边缘) 和横贯结构, 可以通过与某些( 加权) 象子行走模型的对应方式, 通过由凯尼翁、 米勒、 谢菲尔德 和 威尔逊 产生的双极形的合适专业来设置。 然后我们得出准确和无症状的计数结果, 特别是我们证明( 计算 ) $n+2 的平面双极形结构数等于 $( 7/8 美元) 的平面形结构数, 以及 以 $+2 美元 的半径结构数( 大约为 $ > 0美元) 等同的反向估计值 $t_ n\ sim c\\! ( 27/2) ⁇ -\\\\\\\\\\\\ mathrm{arccos}} $( 7/8), 也确保相关生成函数不是 D- finite 。