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A007835 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
1, 1, 3, 8, 21, 52, 124, 284, 629, 1352, 2829, 5777, 11544 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The trees in question might also be called unlabeled acyclic connected digraphs, totally acyclic digraphs or acyclic posets.

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.

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)}.

For n=4 there are 8 directed trees:

O->-O->-O->-O, O->-O->-O-<-O, O-<-O-<-O->-O, O->-O-<-O->-O,

......................

O .... O .... O .... O

| .... | .... | .... |

V .... ^ .... V .... ^

| .... | .... | .... |

O-<--O O->--O O-<--O O->--O

| .... | .... | .... |

^ .... V .... V .... ^

| .... | .... | .... |

O .... O .... O .... O

(see A000238 for the number of them with n nodes). It turns out that all of these give different sets, so a(4)=8.

For n=5 there are 27 directed trees. The following groups of trees give the same set:

O-->O<--O<--O<--O {(0,1), (0,1), (2,0), (1,1), (1,1)}

O-->O-->O<--O<--O {(0,1), (0,1), (2,0), (1,1), (1,1)}

------------------------------------------------------------

O<--O-->O-->O-->O {(1,0), (1,0), (0,2), (1,1), (1,1)}

O<--O<--O-->O-->O {(1,0), (1,0), (0,2), (1,1), (1,1)}

------------------------------------------------------------

O-->O<--O<--O-->O {(0,1), (1,0), (2,0), (1,1), (0,2)}

O-->O-->O<--O-->O {(0,1), (1,0), (2,0), (1,1), (0,2)}

O-->O<--O-->O-->O {(0,1), (1,0), (2,0), (1,1), (0,2)}

------------------------------------------------------------

............O

............|

............v

....O<--O<--O-->O {(0,1), (1,0), (1,0), (1,1), (1,2)}

.............

............O

............^

............|

....O-->O-->O-->O {(0,1), (1,0), (1,0), (1,1), (1,2)}

------------------------------------------------------------

............O

............^

............|

....O-->O-->O<--O {(0,1), (0,1), (1,0), (1,1), (2,1)}

.............

............O

............|

............v

....O<--O<--O<--O {(0,1), (0,1), (1,0), (1,1), (2,1)}

------------------------------------------------------------

There are no other duplications, so a(5)=23, as claimed.

LINKS

Table of n, a(n) for n=1..13.

P. Aubry, Letter to N. J. A. Sloane with attachment, Feb. 1994

Index entries for sequences related to trees

CROSSREFS

Cf. A000238.

Sequence in context: A322059 A259714 A096770 * A152086 A014396 A170881

Adjacent sequences:  A007832 A007833 A007834 * A007836 A007837 A007838

KEYWORD

nonn,more

AUTHOR

Philippe Aubry (philippe.aubry(AT)oncfs.gouv.fr), Oct 02 1994

EXTENSIONS

Edited by N. J. A. Sloane, May 17 2006

a(12)-a(13) from and example in comment clarified by Sean A. Irvine, Feb 04 2018

STATUS

approved

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Last modified September 21 00:17 EDT 2019. Contains 327252 sequences. (Running on oeis4.)