%I
%S 1,1,3,9,33,131,561,2535,11971,58579,295297,1526427,8061879,43380351,
%T 237266225,1316536991,7399318871,42065753191,241628448517,
%U 1400957386207
%N Number of increasing planar stgraphs with n edges.
%C From _Michael Wallner_, Aug 17 2020: (Start)
%C A graph with m vertices is called increasing, or topolicically ordered, if each vertex v is assigned a number l(v) from {1,...,m} and if there is a directed path from vertex v to vertex w then l(v) < l(w).
%C Note that a planar stgraph is an actual embedding into the plane, i.e., we distinguish a lefttoright order on the children of each node. In other words, it is a plane graph, a concept also sometimes called isotopy. (End)
%D Ioannis G. Tollis, Giuseppe Di Battista, Peter Eades and Roberto Tamassia, Graph drawing, Prentice Hall Inc., Upper Saddle River, NJ, 1999. Algorithms for the visualization of graphs. MR2064104 (2005i:68067). See Sect. 4.2.
%H Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, <a href="https://arxiv.org/abs/1911.10288">On sequences associated to the invariant theory of rank two simple Lie algebras</a>, arXiv:1911.10288 [math.CO], 2019.
%H James Cranch, <a href="http://jdc41.user.srcf.net/research/pasting/Pasting.pdf">Representing and Enumerating TwoDimensional Pasting Diagrams</a>, 2014.
%F Conjecture: (n+2)*(n+3)*a(n) = 2*n*(2*n+1)*a(n1) + (n2)*(19*n9)*a(n2) + 14*(n3)*(n2)*a(n3).  _Vaclav Kotesovec_, Feb 14 2014
%e From _Michael Wallner_, Aug 17 2020: (Start)
%e All edges are directed to the bottom.
%e a(3)=3:
%e 1 1 1
%e  \ /
%e 2  2 2 
%e  / \
%e 3 3 3
%e 
%e 4
%e a(4)=9:
%e 1 1 1 1 1 1 1 1 1
%e    \ / \ / / \ / \
%e 2 2 2 2   2 2   2 2 3 3 2
%e  \ / / \     \ / \ /
%e 3 3   3 3 3 3   3 4 4
%e  / \   / \
%e 4 4 4 4 4 4 4
%e 
%e 5
%e (End)
%K nonn,more
%O 1,3
%A _N. J. A. Sloane_, Feb 03 2014
%E Name clarified by _Michael Wallner_, Aug 17 2020
