A058876 -id:A058876 - cp's OEIS frontend

A058877 Number of labeled acyclic digraphs with n nodes containing exactly n-1 points of in-degree zero.

0, 2, 9, 28, 75, 186, 441, 1016, 2295, 5110, 11253, 24564, 53235, 114674, 245745, 524272, 1114095, 2359278, 4980717, 10485740, 22020075, 46137322, 96468969, 201326568, 419430375, 872415206, 1811939301, 3758096356, 7784628195, 16106127330, 33285996513

Offset: 1

N. J. A. Sloane, Jan 07 2001

Convolution of 2^n+1 (A000051) and 2^n-1 (A000225). - Graeme McRae, Jun 07 2006
Let Q be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all nonempty elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |Q|. - Ross La Haye, Feb 20 2008, Oct 21 2008
Also: convolution of A006589 with A000012 (i.e., partial sums of A006589). - R. J. Mathar, Jan 25 2009
The La Haye binary relation Q is more clearly stated as x is nonempty and y has one more element than x. If x is a k-set than the number of such pairs is binomial( n, k) * (n-k). - Michael Somos, Mar 29 2012
Select one of the n nodes of the digraph and select a nonempty subset of the rest to connect to the selected node. This can be done in n * (2^(n-1) - 1) ways. - Michael Somos, Mar 29 2012
Column 1 of A198204. - Peter Bala, Aug 01 2012
a(n) is the number of ternary sequences of length n that contain one 0 and at least one 1.  For example, a(3)=9 since the sequences are the 3 permutations of 011 and the 6 permutations of 012. - Enrique Navarrete, Apr 05 2021
a(n) is also the number of multiplications required to compute the permanent of general n X n matrices using canonical trellis method (see Theorem 5, p. 10 in Kiah et al.). - Stefano Spezia, Nov 02 2021

			G.f. = 2*x^2 + 9*x^3 + 28*x^4 + 75*x^5 + 186*x^6 + 441*x^7 + 1016*x^8 + ...

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, (1.6.4).
Gerta Rucker and Christoph Rucker, "Walk counts, Labyrinthicity and complexity of acyclic and cyclic graphs and molecules", J. Chem. Inf. Comput. Sci., 40 (2000), 99-106. See Table 1 on page 101. [From Parthasarathy Nambi, Sep 26 2008]

Vincenzo Librandi, Table of n, a(n) for n = 1..2001
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 12.
Han Mao Kiah, Alexander Vardy and Hanwen Yao, Computing Permanents on a Trellis, arXiv:2107.07377 [cs.IT], 2021.
Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017), remark 3.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 36.
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Second column of A058876. Cf. A003025, A003026.
Column k=1 of A133399.
Cf. A198204.

[(n+1)*2^n-n-1: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011

[seq (stirling2(n,2)*n,n=1..29)]; # Zerinvary Lajos, Dec 06 2006
a:=n->sum(k*binomial(n,k), k=2..n): seq(a(n), n=1..29); # Zerinvary Lajos, May 08 2007

Table[(n+1)*2^n-n-1, {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 30 2011 *)
a[ n_] := Sum[ Binomial[ n, k] (n - k), {k, n-1}]; (* Michael Somos, Mar 29 2012 *)

{a(n) = if( n<1, 0, n * (2^(n-1) - 1))} /* Michael Somos, Mar 29 2012 */

[stirling_number2(i,2)*i for i in range(1,26)] # Zerinvary Lajos, Jun 27 2008

[(n+1)*gaussian_binomial(n,1,2) for n in range(0,29)] # Zerinvary Lajos, May 31 2009

a(n+1) = (n+1)*2^n - n - 1 = Sum_{j=0..n} (n+j)*2^(n-j-1) = A048493(n)-1 = Column sum of A062111. - Henry Bottomley, May 30 2001
From R. J. Mathar, Jan 25 2009: (Start)
G.f.: x^2*(2-3*x)/((1-2*x)*(1-x))^2.
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4). (End)
a(n) = Sum_{k=1..n-1} binomial(n, k) * (n-k). - Michael Somos, Mar 29 2012
E.g.f: x*exp(x)*(exp(x)-1). - Enrique Navarrete, Apr 05 2021

More terms from Vladeta Jovovic, Apr 10 2001

A003025 Number of n-node labeled acyclic digraphs with 1 out-point.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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)

			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)

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).

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.

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.

Table[Length[Select[Subsets[Subsets[Range[n]],{n}],Count[#,{}]==1&&Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,4}] (* _Gus Wiseman, Jan 02 2024 *)

\\ requires A058876.
my(T=A058876(20)); vector(#T, n, T[n][1]) \\ Andrew Howroyd, Dec 27 2021

\\ 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

a(n) = (-1)^(n-1) * n * A134531(n). - Gus Wiseman, Jan 02 2024

More terms from Vladeta Jovovic, Apr 10 2001

A122078 Triangle read by rows: T(n,k) is the number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 11, 16, 0, 1, 4, 25, 108, 164, 0, 1, 5, 47, 422, 2168, 3341, 0, 1, 6, 78, 1251, 15484, 88747, 138101, 0, 1, 7, 120, 3124, 79836, 1215783, 7409117, 11578037, 0, 1, 8, 174, 6925, 333004, 11620961, 199203464, 1252610909, 1961162564, 0

Offset: 0

N. J. A. Sloane, Oct 18 2006

			Triangle T(n,k) begins:
  1:
  1, 0;
  1, 1,  0;
  1, 2,  3,    0;
  1, 3, 11,   16,     0;
  1, 4, 25,  108,   164,     0;
  1, 5, 47,  422,  2168,  3341,      0;
  1, 6, 78, 1251, 15484, 88747, 138101, 0;
  ...

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Andrew Howroyd, Table of n, a(n) for n = 0..495 (rows 0..30; rows 0..15 from R. W. Robinson)
Andrew Howroyd, PARI program, Dec 2021, updated Jan 2022.

Row sums give A003087.
Diagonals include A000007, A350415.
Cf. A058876 (labeled case), A350447, A350448, A350449, A350450.

\\ See link for program code.
{ my(T=AcyclicDigraphsByNonSources(8)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 31 2021

Zero terms inserted by Andrew Howroyd, Dec 29 2021

A003026 Number of n-node labeled acyclic digraphs with 2 out-points.

Original entry on oeis.org

1, 9, 198, 10710, 1384335, 416990763, 286992935964, 444374705175516, 1528973599758889005, 11573608032229769067465, 191141381932394665770442818, 6839625961762363728765713227698

Offset: 2

N. J. A. Sloane

nonn

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).

Andrew Howroyd, Table of n, a(n) for n = 2..50

A diagonal of A058876.

Cf. A003025.

\\ requires A058876.
my(T=A058876(20)); vector(#T-1, n, T[n+1][2]) \\ Andrew Howroyd, Dec 27 2021

More terms from Vladeta Jovovic, Apr 10 2001

A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 15, 9, 1, 0, 316, 198, 28, 1, 0, 16885, 10710, 1610, 75, 1, 0, 2174586, 1384335, 211820, 10575, 186, 1, 0, 654313415, 416990763, 64144675, 3268125, 61845, 441, 1, 0, 450179768312, 286992935964, 44218682312, 2266772550, 43832264, 336924, 1016, 1

Offset: 0

Geoffrey Critzer, Apr 02 2023

Also the number of sets of n nonempty subsets of {1..n}, k of which are singletons, such that there is only one way to choose a different element from each. For example, row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{3},{1,2}}
  {{2},{1,2},{1,3}}    {{1},{3},{2,3}}
  {{2},{1,2},{2,3}}    {{2},{3},{1,2}}
  {{2},{1,3},{2,3}}    {{2},{3},{1,3}}
  {{3},{1,2},{1,3}}    {{1},{2},{1,2,3}}
  {{3},{1,2},{2,3}}    {{1},{3},{1,2,3}}
  {{3},{1,3},{2,3}}    {{2},{3},{1,2,3}}
  {{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}}

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,    15,     9,    1;
  0,   316,   198,   28,  1;
  0, 16885, 10710, 1610, 75, 1;
  ...

E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

Cf. A058876 (mirror), A361579, A224069.
Row-sums are A003024, unlabeled A003087.
Column k = 1 is A003025(n) = |n*A134531(n)|.
Column k = n-1 is A058877.
For fixed sinks we get A368602.
A058891 counts set-systems, unlabeled A000612.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A059201, A082402, A088957, A133686, A334282, A350415, A367904, A367908, A368600, A368601.

nn = 8; B[n_] := n! 2^Binomial[n, 2] ;ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /. Table[z^i -> z^i/2^Binomial[i, 2], {i, 0, nn}];Table[Take[(Table[B[n], {n, 0, nn}] CoefficientList[ Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-z]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}] // Grid
nv=4;Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]],{n,0,nv},{k,0,n}]

T(n,k) = A368602(n,k) * binomial(n,k). - Gus Wiseman, Jan 03 2024

A060335 Number of n-node labeled acyclic digraphs with 3 out-points.

Original entry on oeis.org

1, 28, 1610, 211820, 64144675, 44218682312, 68501035223124, 235728863806525320, 1784437537982029455525, 29470895991194487015464740, 1054563682428338672254476697886, 81276604641664521211218527866093204

Offset: 3

Vladeta Jovovic, Apr 10 2001

nonn

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.

Andrew Howroyd, Table of n, a(n) for n = 3..50

A diagonal of A058876.

Cf. A003025, A003026.

\\ requires A058876.
my(T=A058876(20)); vector(#T-2, n, T[n+2][3]) \\ Andrew Howroyd, Dec 27 2021

A060337 Number of labeled acyclic digraphs with n nodes containing exactly n-2 points of in-degree zero.

Original entry on oeis.org

15, 198, 1610, 10575, 61845, 336924, 1751076, 8801325, 43141175, 207347778, 980828238, 4578689115, 21135851625, 96628899960, 438068838536, 1971349880985, 8813183238315, 39169902510270, 173172640973010

Offset: 3

Vladeta Jovovic, Apr 10 2001

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, (1.6.4).
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.

Andrew Howroyd, Table of n, a(n) for n = 3..500
Index entries for linear recurrences with constant coefficients, signature (21,-189,955,-2982,5964,-7640,6048,-2688,512).

Third column of A058876.

Cf. A003025, A003026.

LinearRecurrence[{21,-189,955,-2982,5964,-7640,6048,-2688,512},{15,198,1610,10575,61845,336924,1751076,8801325,43141175},20] (* Harvey P. Dale, Mar 23 2022 *)

\\ requires A058876.
my(T=A058876(25)); vector(#T-2, n, T[n+2][n]) \\ Andrew Howroyd, Dec 27 2021

G.f.: x^3*(15 - 117*x + 287*x^2 - 138*x^3 - 300*x^4 + 280*x^5)/((1 - x)*(1 - 2*x)*(1 - 4*x))^3. - Andrew Howroyd, Dec 27 2021

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A058877 Number of labeled acyclic digraphs with n nodes containing exactly n-1 points of in-degree zero.

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A003025 Number of n-node labeled acyclic digraphs with 1 out-point.

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A122078 Triangle read by rows: T(n,k) is the number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes (0 <= k <= n).

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A003026 Number of n-node labeled acyclic digraphs with 2 out-points.

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A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

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A060335 Number of n-node labeled acyclic digraphs with 3 out-points.

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PARI

A060337 Number of labeled acyclic digraphs with n nodes containing exactly n-2 points of in-degree zero.

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