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%I #22 Feb 05 2018 03:20:24
%S 1,1,3,8,21,52,124,284,629,1352,2829,5777,11544
%N Number of unordered sets of pairs (in-degree, out-degree) for nodes of directed trees on n unlabeled nodes (the edges are directed in arbitrary directions, the tree is unrooted).
%C The trees in question might also be called unlabeled acyclic connected digraphs, totally acyclic digraphs or acyclic posets.
%C Comments from _Dean Hickerson_, May 17 2006: For each directed tree with n nodes, write down the set of pairs (in-degree, out-degree) that occur in the tree. Then count how many different sets you get that way.
%C For n=3 there are 3 such sets, namely: O-->O-->O {(0,1), (1,0), (1,1)}, O-->O<--O {(0,1), (2,0)}, O<--O-->O {(1,0), (0,2)}.
%C For n=4 there are 8 directed trees:
%C O->-O->-O->-O, O->-O->-O-<-O, O-<-O-<-O->-O, O->-O-<-O->-O,
%C ......................
%C O .... O .... O .... O
%C | .... | .... | .... |
%C V .... ^ .... V .... ^
%C | .... | .... | .... |
%C O-<--O O->--O O-<--O O->--O
%C | .... | .... | .... |
%C ^ .... V .... V .... ^
%C | .... | .... | .... |
%C O .... O .... O .... O
%C (see A000238 for the number of them with n nodes). It turns out that all of these give different sets, so a(4)=8.
%C For n=5 there are 27 directed trees. The following groups of trees give the same set:
%C O-->O<--O<--O<--O {(0,1), (0,1), (2,0), (1,1), (1,1)}
%C O-->O-->O<--O<--O {(0,1), (0,1), (2,0), (1,1), (1,1)}
%C ------------------------------------------------------------
%C O<--O-->O-->O-->O {(1,0), (1,0), (0,2), (1,1), (1,1)}
%C O<--O<--O-->O-->O {(1,0), (1,0), (0,2), (1,1), (1,1)}
%C ------------------------------------------------------------
%C O-->O<--O<--O-->O {(0,1), (1,0), (2,0), (1,1), (0,2)}
%C O-->O-->O<--O-->O {(0,1), (1,0), (2,0), (1,1), (0,2)}
%C O-->O<--O-->O-->O {(0,1), (1,0), (2,0), (1,1), (0,2)}
%C ------------------------------------------------------------
%C ............O
%C ............|
%C ............v
%C ....O<--O<--O-->O {(0,1), (1,0), (1,0), (1,1), (1,2)}
%C .............
%C ............O
%C ............^
%C ............|
%C ....O-->O-->O-->O {(0,1), (1,0), (1,0), (1,1), (1,2)}
%C ------------------------------------------------------------
%C ............O
%C ............^
%C ............|
%C ....O-->O-->O<--O {(0,1), (0,1), (1,0), (1,1), (2,1)}
%C .............
%C ............O
%C ............|
%C ............v
%C ....O<--O<--O<--O {(0,1), (0,1), (1,0), (1,1), (2,1)}
%C ------------------------------------------------------------
%C There are no other duplications, so a(5)=23, as claimed.
%H P. Aubry, <a href="/A007835/a007835.pdf">Letter to N. J. A. Sloane with attachment, Feb. 1994</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%Y Cf. A000238.
%K nonn,more
%O 1,3
%A Philippe Aubry (philippe.aubry(AT)oncfs.gouv.fr), Oct 02 1994
%E Edited by _N. J. A. Sloane_, May 17 2006
%E a(12)-a(13) from and example in comment clarified by _Sean A. Irvine_, Feb 04 2018
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