A003025 - OEIS

A003025

Number of n-node labeled acyclic digraphs with 1 out-point.
(Formerly M2083)

1, 2, 15, 316, 16885, 2174586, 654313415, 450179768312, 696979588034313, 2398044825254021110, 18151895792052235541515, 299782788128536523836784628, 10727139906233315197412684689421

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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

Andrew Howroyd, Table of n, a(n) for n = 1..50

Marcel et al., Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?, MathOverflow, 2021.

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.

Cf. A003026, A003028, A345258.

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.

Cf. A000169, A058891, A059201, A082402, A088957, A323818, A334282, A367904, A368600, A368601.

Sequence in context: A304120 A255929 A059167 * A015200 A331344 A030642

Adjacent sequences: A003022 A003023 A003024 * A003026 A003027 A003028