A081064 -id:A081064 - cp's OEIS frontend

A147964 Number of consistent sets of 10 irreflexive binary order relationships over n objects.

120, 691020, 128047374, 6519340912, 156097542888, 2259242749800, 22815705739244, 175939638868224, 1099964624581680, 5812510584460580, 26753072198342490, 109684475412107232, 407515671392921520, 1390695205822539984, 4406577363489470616, 13079027432832437440

Offset: 5

R. H. Hardin, May 04 2009

V. I. Rodionov, On the number of labeled acyclic digraphs, Discr. Math. 105 (1-3) (1992), 319-321.

Related sequences for the number of consistent sets of k irreflexive binary order relationships over n objects: A147796 (k = 3), A147817 (k = 4), A147821 (k = 5), A147860 (k = 6), A147872 (k = 7), A147881 (k = 8), A147883 (k = 9).

Column k = 10 of A081064.

Table[(n - 4)*(n - 3)*(n - 2)*(n - 1)*n*(n^15 - 80*n^13 - 300*n^12 + 1366*n^11 + 18300*n^10 + 117700*n^9 + 293220*n^8 - 4873571*n^7 - 63731100*n^6 - 168619940*n^5 + 2528179320*n^4 + 17989477164*n^3 - 56994404400*n^2 - 561199055760*n + 1856094609600)/3628800, {n, 5, 20}] (* Wesley Ivan Hurt, Apr 12 2020 *)

a(n) = (n-4)*(n-3)*(n-2)*(n-1)*n*(n^15 - 80*n^13 - 300*n^12 + 1366*n^11 + 18300*n^10 + 117700*n^9 + 293220*n^8 - 4873571*n^7 - 63731100*n^6 - 168619940*n^5 + 2528179320*n^4 + 17989477164*n^3 - 56994404400*n^2 - 561199055760*n + 1856094609600)/3628800. - Vaclav Kotesovec, Apr 11 2020

More terms from Vaclav Kotesovec, Apr 11 2020

Offset changed to n=5 by Petros Hadjicostas, Apr 11 2020

A342587 Triangle, read by rows: T(n,k) is the number of labeled order relations on n nodes in which the longest chain has k nodes (n>=1, 1<=k<=n).

Original entry on oeis.org

1, 1, 2, 1, 12, 6, 1, 86, 108, 24, 1, 840, 2310, 960, 120, 1, 11642, 65700, 42960, 9000, 720, 1, 227892, 2583126, 2510760, 712320, 90720, 5040, 1, 6285806, 142259628, 199357704, 71310960, 11481120, 987840, 40320, 1, 243593040, 11012710470, 21774014640, 9501062760, 1781015040

Offset: 1

R. J. Mathar and Brendan McKay, Mar 16 2021

Corrects Comtet's table for k=4 and 5 in row n=8.

			Triangle T(n,k) (with n >= 1 and 1 <= k <= n) begins as follows:
  1;
  1,      2;
  1,     12,       6;
  1,     86,     108,      24;
  1,    840,    2310,     960,    120;
  1,  11642,   65700,   42960,   9000,   720;
  1, 227892, 2583126, 2510760, 712320, 90720, 5040;
  ...

Brendan McKay, Table of T(n,k) for n = 1..13
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

Cf. A000142 (diagonal), A001035 (row sums), A055531 (k=2), A055532 (k=3), A055533 (subdiagonal), A055534 (subdiagonal), A081064, A342501 (connected).

A350447 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n unlabeled nodes with k arcs, n >=0, k = 0..(n-1)*n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 9, 9, 6, 1, 1, 1, 4, 12, 37, 60, 80, 63, 33, 10, 1, 1, 1, 4, 13, 51, 163, 407, 796, 1169, 1291, 1057, 649, 281, 85, 15, 1, 1, 1, 4, 13, 54, 215, 846, 2690, 7253, 15703, 27596, 39057, 44902, 41723, 31336, 18844, 8983, 3325, 920, 180, 21, 1

Offset: 0

Andrew Howroyd, Dec 31 2021

			Triangle begins:
  [0] 1;
  [1] 1;
  [2] 1, 1;
  [3] 1, 1, 3,  1;
  [4] 1, 1, 4,  9,  9,  6,  1;
  [5] 1, 1, 4, 12, 37, 60, 80, 63, 33, 10, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

The labeled version is A081064.
Row sums are A003087.
Cf. A122078, A350449.

\\ See PARI link in A122078 for program code.
{ my(T=AcyclicDigraphsByArcs(6)); for(n=1, #T, print(T[n])) }

A350487 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n labeled nodes with k arcs and a global source, n >= 1, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 2, 0, 0, 9, 6, 0, 0, 0, 64, 132, 96, 24, 0, 0, 0, 0, 625, 2640, 4850, 4900, 2850, 900, 120, 0, 0, 0, 0, 0, 7776, 55800, 186480, 379170, 516660, 491040, 328680, 152640, 46980, 8640, 720, 0, 0, 0, 0, 0, 0, 117649, 1286670, 6756120, 22466010

Offset: 1

Andrew Howroyd, Jan 01 2022

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 9,  6;
  [4] 0, 0, 0, 64, 132,   96,   24;
  [5] 0, 0, 0,  0, 625, 2640, 4850, 4900, 2850, 900, 120;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)
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.

Row sums are A003025.
Leading diagonal is A000169.
The unlabeled version is A350488.
Cf. A081064.

T(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)*((1+'y)^(n-k)-1)^k*a[n-k])); [Vecrev(p) | p <- a]}
{ my(A=T(6)); for(n=1, #A, print(A[n])) }

A350909 Triangle read by rows: T(n,k) is the number of weakly connected acyclic digraphs on n labeled nodes with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 2, 0, 0, 12, 6, 0, 0, 0, 128, 186, 108, 24, 0, 0, 0, 0, 2000, 5640, 7840, 6540, 3330, 960, 120, 0, 0, 0, 0, 0, 41472, 189480, 456720, 730830, 832370, 690300, 416160, 178230, 51480, 9000, 720, 0, 0, 0, 0, 0, 0, 1075648, 7178640, 26035800, 65339820

Offset: 1

Andrew Howroyd, Jan 29 2022

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 12,   6;
  [4] 0, 0,  0, 128,  186,  108,   24;
  [5] 0, 0,  0,   0, 2000, 5640, 7840, 6540, 3330, 960, 120;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)

Row sums are A082402.
Leading diagonal is A097629.
The unlabeled version is A350449.
Cf. A057273, A062735, A081064.

G(n)={my(v=vector(n+1)); v[1]=1; for(n=1, n, v[n+1]=sum(k=1, n, -(-1)^k*(1+y)^(k*(n-k))*v[n-k+1]/k!))/n!; Ser(v)}
row(n)={Vecrev(n!*polcoef(log(G(n)), n))}
{ for(n=1, 6, print(row(n))) }

A008285 Erroneous version of A342587.

Original entry on oeis.org

1, 1, 2, 1, 12, 6, 1, 86, 108, 24, 1, 840, 2310, 960, 120, 1, 11642, 65700, 42960, 9000, 720, 1, 227892, 2583126, 2510760, 712320, 90720, 5040, 1, 6285806, 142259628, 199424904, 71243760, 11481120, 987840, 40320

Offset: 1

N. J. A. Sloane

dead

			Triangle T(n,k) (with n >= 1 and 1 <= k <= n) begins as follows:
  1;
  1,      2;
  1,     12,       6;
  1,     86,     108,      24;
  1,    840,    2310,     960,    120;
  1,  11642,   65700,   42960,   9000,   720;
  1, 227892, 2583126, 2510760, 712320, 90720, 5040;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

Cf. A000142 (diagonal), A001035 (row sums), A055531 (k=2), A055532 (k=3), A055533 (subdiagonal), A081064, A342501 (connected).

cp's OEIS Frontend

A147964 Number of consistent sets of 10 irreflexive binary order relationships over n objects.

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A342587 Triangle, read by rows: T(n,k) is the number of labeled order relations on n nodes in which the longest chain has k nodes (n>=1, 1<=k<=n).

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A350447 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n unlabeled nodes with k arcs, n >=0, k = 0..(n-1)*n/2.

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A350487 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n labeled nodes with k arcs and a global source, n >= 1, k = 0..n*(n-1)/2.

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A350909 Triangle read by rows: T(n,k) is the number of weakly connected acyclic digraphs on n labeled nodes with k arcs, k=0..n*(n-1).

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A008285 Erroneous version of A342587.

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