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A236408 Number of increasing planar st-graphs with n edges. 0

%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 st-graphs 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 st-graph is an actual embedding into the plane, i.e., we distinguish a left-to-right 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 Two-Dimensional Pasting Diagrams</a>, 2014.

%F Conjecture: (n+2)*(n+3)*a(n) = 2*n*(2*n+1)*a(n-1) + (n-2)*(19*n-9)*a(n-2) + 14*(n-3)*(n-2)*a(n-3). - _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

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Last modified July 23 19:00 EDT 2021. Contains 346259 sequences. (Running on oeis4.)