2024/08/10 by Keiichi Shigechi, Shigechi, Keiichi
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Combinatorial Mathematics #Computational Geometry and Mesh Generation
paper · pdf · doi:10.48550/arxiv.2408.05483
We study a partially ordered set of planar labeled rooted trees by use of combinatorial objects called Dyck tilings. A generating function of the poset is factorized when the minimum element of the poset is 312-avoiding and satisfies some extra condition. We define a cover relation on rational Dyck tilings by that of labeled trees, and show that increasing and decreasing labelings are dual to each other. We consider two decompositions of a rational (a,b)-Dyck tiling: one is into ab Dyck tilings and the other is into a (1,b)-Dyck tilings. In the first case, we show that the weight of the (a,b)-Dyck tiling is the sum of the weights of ab Dyck tilings. In the second case, we introduce a cover relation on (1,b)-Dyck tilings and obtain a poset of (a,b)-Dyck tilings by this decomposition.