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A090376 Number of rooted generalized quadrangular dissections of weight n of a closed disk: planar maps having the external face bounded by a polygon and all internal faces of size 4. 1

%I

%S 1,4,15,80,362,1832,8994,46384,238838,1257824

%N Number of rooted generalized quadrangular dissections of weight n of a closed disk: planar maps having the external face bounded by a polygon and all internal faces of size 4.

%C Some boundary mutually non-adjacent nodes of valency 2 are marked as singular; (boundary) edges incident to them are also called singular. The maps are considered up to rotations and reflections. Rooting means distinguishing a non-singular edge, an end and an internal side of it. n is the number of internal edges plus half of the number of non-singular boundary edges.

%C No formula is known. For any generalized quadrangular dissection, s==n (mod 2), where s is the number of singular nodes.

%H V. A. Liskovets, <a href="http://dx.doi.org/10.1016/0012-365X(94)00347-L">A reductive technique for enumerating nonisomorphic planar maps</a>, Discr. Math., v.156 (1996), 197-217.

%e The four rooted generalized quadrangular dissections of weight 1 are

%e ...................____......____..

%e .X<---X..X---<X.../....\..../....\.

%e .|....|..|....|..X<--X..O..X--<X..O

%e .|....|..|....|...\____/....\____/.

%e .X----O..X----O....................

%e where O is the singular node and -> is the rooted edge-end.

%Y Cf. A006385.

%K nonn,more

%O 0,2

%A _Valery A. Liskovets_, Dec 03 2003

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Last modified October 1 17:45 EDT 2020. Contains 337444 sequences. (Running on oeis4.)