论文标题
平面图和平面线性正常$λ$ - 具有连通性条件的双线
Bijections between planar maps and planar linear normal $λ$-terms with connectivity condition
论文作者
论文摘要
线性$λ$ terms的枚举最近引起了很多关注，部分原因是它们与组合图的链接。 Zeilberger and Giorgetti（2015）在平面线性$λ$ - terms和Planar Maps之间进行了递归培训，当将其仅限于2个连接的$λ$ terms（即没有封闭的子函数）时，它们会导致无用的无用平面图。受这一限制的启发，Zeilberger and Reed（2019）猜想3连接的平面线性正常$λ$ terms具有与两部分平面图相同的计数公式。在本文中，我们通过在这两个家庭之间进行直接培养来解决这一猜想。此外，使用类似的方法，我们在平面线性正常$λ$ terms和Planar Maps之间进行了直接的培养，其限制到2相连的$λ$ terms会导致无环平面地图。即使在二倍地图上，这种培养物似乎与Zeilberger和Giorgetti的培训不同。我们还探索了我们的列举后果。
The enumeration of linear $λ$-terms has attracted quite some attention recently, partly due to their link to combinatorial maps. Zeilberger and Giorgetti (2015) gave a recursive bijection between planar linear normal $λ$-terms and planar maps, which, when restricted to 2-connected $λ$-terms (i.e., without closed sub-terms), leads to bridgeless planar maps. Inspired by this restriction, Zeilberger and Reed (2019) conjectured that 3-connected planar linear normal $λ$-terms have the same counting formula as bipartite planar maps. In this article, we settle this conjecture by giving a direct bijection between these two families. Furthermore, using a similar approach, we give a direct bijection between planar linear normal $λ$-terms and planar maps, whose restriction to 2-connected $λ$-terms leads to loopless planar maps. This bijection seems different from that of Zeilberger and Giorgetti, even after taking the map dual. We also explore enumerative consequences of our bijections.
