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A003025 Number of n-node labeled acyclic digraphs with 1 out-point.
(Formerly M2083)
16
1, 2, 15, 316, 16885, 2174586, 654313415, 450179768312, 696979588034313, 2398044825254021110, 18151895792052235541515, 299782788128536523836784628, 10727139906233315197412684689421 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
From Gus Wiseman, Jan 02 2024: (Start)
Also the number of n-element sets of finite nonempty subsets of {1..n}, including a unique singleton, such that there is exactly one way to choose a different element from each. For example, the a(0) = 0 through a(3) = 15 set-systems are:
. {{1}} {{1},{1,2}} {{1},{1,2},{1,3}}
{{2},{1,2}} {{1},{1,2},{2,3}}
{{1},{1,3},{2,3}}
{{2},{1,2},{1,3}}
{{2},{1,2},{2,3}}
{{2},{1,3},{2,3}}
{{3},{1,2},{1,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1},{1,2},{1,2,3}}
{{1},{1,3},{1,2,3}}
{{2},{1,2},{1,2,3}}
{{2},{2,3},{1,2,3}}
{{3},{1,3},{1,2,3}}
{{3},{2,3},{1,2,3}}
These set-systems are all connected.
The case of labeled graphs is A000169.
(End)
REFERENCES
R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
FORMULA
a(n) = (-1)^(n-1) * n * A134531(n). - Gus Wiseman, Jan 02 2024
EXAMPLE
a(2) = 2: o-->--o (2 ways)
a(3) = 15: o-->--o-->--o (6 ways) and
o ... o o-->--o
.\ . / . \ . /
. v v ... v v
.. o ..... o
(3 ways) (6 ways)
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n]], {n}], Count[#, {_}]==1&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]], {n, 0, 4}] (* Gus Wiseman, Jan 02 2024 *)
PROG
(PARI) \\ requires A058876.
my(T=A058876(20)); vector(#T, n, T[n][1]) \\ Andrew Howroyd, Dec 27 2021
(PARI) \\ see Marcel et al. link (formula for a').
seq(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1) * binomial(n, k) * (2^(n-k)-1)^k * a[n-k])); a} \\ Andrew Howroyd, Jan 01 2022
CROSSREFS
A diagonal of A058876.
Row sums of A350487.
The unlabeled version is A350415.
Column k=1 of A361718.
For any number of sinks we have A003024, unlabeled A003087.
For n-1 sinks we have A058877.
For a fixed sink we have A134531 (up to sign), column k=1 of A368602.
Sequence in context: A304120 A255929 A059167 * A015200 A331344 A030642
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 10 2001
STATUS
approved

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)