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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).
1

%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