cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A370961 a(n) = number of acyclic orientations of the complete tripartite graph K_{n,n,n}.

Original entry on oeis.org

1, 6, 426, 122190, 90768378, 138779942046, 379578822373866, 1689637343582548590, 11434884125767376107098, 111765072808554847704145086, 1515592947854931941485836600906, 27609710924806869786487193747541390, 658043992934027491354757341987635993018
Offset: 0

Views

Author

N. J. A. Sloane, Apr 04 2024

Keywords

Links

Crossrefs

Cf. A212220, A266858, A267383, A370960.
Main diagonal of A372254.
Row n=3 of A372326.

Formula

a(n) = A266858(3n) = A267383(3n,3). - Alois P. Heinz, Apr 17 2024
a(n) = Sum_{k=0..3n} (-1)^k * A212220(n,k). - Alois P. Heinz, May 02 2024

Extensions

More terms from Don Knuth, Apr 07 2024
a(0)=1 prepended by Alois P. Heinz, Apr 17 2024

A372261 Number T(n,k,j) of acyclic orientations of the complete tripartite graph K_{n,k,j}; triangle of triangles T(n,k,j), n>=0, k=0..n, j=0..k, read by rows.

Original entry on oeis.org

1, 1, 2, 6, 1, 4, 18, 14, 78, 426, 1, 8, 54, 46, 330, 2286, 230, 1902, 15402, 122190, 1, 16, 162, 146, 1374, 12090, 1066, 10554, 101502, 951546, 6902, 76110, 822954, 8724078, 90768378, 1, 32, 486, 454, 5658, 63198, 4718, 57054, 657210, 7290942, 41506, 525642, 6495534, 78463434, 928340190
Offset: 0

Views

Author

Alois P. Heinz, Apr 24 2024

Keywords

Comments

An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.

Examples

			Triangle of triangles T(n,k,j) begins:
    1;
  ;
    1;
    2,    6;
  ;
    1;
    4,   18;
   14,   78,   426;
  ;
    1;
    8,   54;
   46,  330,  2286;
  230, 1902, 15402, 122190;
  ;
  ...

Links

Crossrefs

T(n,k,0) for k=0..9 give: A000012, A000079, A027649, A027650, A027651, A283811, A283812, A283813, A284032, A284033.
T(n,n,n) gives A370961.
T(n,n,0) gives A048163(n+1).
T(n+1,n,0) gives A188634(n+1).
T(n,1,1) gives A008776.
T(n,2,2) gives A370960.
Cf. A099594, A212220, A267383, A372254.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(
      expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
    end:
T:= proc() option remember; local q, l, b; q, l, b:= -1, [args],
      proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*
        (q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))
      end; abs(b(0, nops(l)))
    end:
seq(seq(seq(T(n, k, j), j=0..k), k=0..n), n=0..5);
  • Mathematica
    g[n_] := g[n] = If[n == 0, 1, Sum[Expand[x*g[n - j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_, k_, j_] := T[n, k, j] = Module[{q = -1, l = {n, k, j}, b},
   b[n0_, j0_] := b[n0, j0] = If[j0 == 1, Product[q - i, {i, 0, n0 - 1}]*
   (q - n0)^n, Sum[b[n0 + m, j0 - 1]*Coefficient[g[l[[j0]]], x, m],
   {m, 0, l[[j0]]}]];
Abs[b[0, 3]]];
Table[Table[Table[T[n, k, j], {j, 0, k}], {k, 0, n}], {n, 0, 5}] // Flatten (* Jean-François Alcover, Jun 14 2024, after Alois P. Heinz *)

A370960 a(n) = number of acyclic orientations of the complete tripartite graph K_{2,2,n}.

Original entry on oeis.org

14, 78, 426, 2286, 12090, 63198, 327306, 1682766, 8601690, 43768638, 221910186, 1121897646, 5659111290, 28494757278, 143272715466, 719565670926, 3610655860890, 18104646725118, 90728875495146, 454467461514606, 2275631193410490, 11391336159448158, 57009415513961226, 285258058278100686, 1427134339747920090
Offset: 0

Views

Author

N. J. A. Sloane, Apr 04 2024

Keywords

Links

Crossrefs

Cf. A370961.
Row n=2 of A372254.

Extensions

Further terms from Don Knuth, Apr 07 2024
a(0)=14 prepended by Alois P. Heinz, Apr 17 2024

A370613 Total number of acyclic orientations in all complete multipartite graphs K_lambda, where lambda ranges over all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 5, 9, 63, 509, 2959, 22453, 247949, 3080991, 28988331, 407320739, 5122243495, 82583577967, 1430027615585, 22556817627789, 395098668828675, 7979894546677853, 154786744386253387, 3355612019167352821, 78865333300205585345, 1769663675666499515751
Offset: 0

Views

Author

Alois P. Heinz, Apr 30 2024

Keywords

Comments

An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.
All terms are odd.

Links

Crossrefs

Row sums of A370614.
Cf. A000009, A267383, A372254, A372261, A372326, A372395.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(
      expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
    end:
b:= proc(t, n, i) option remember; `if`(i*(i+1)/2 abs(b(0, n$2)):
seq(a(n), n=0..22);

A370614 Triangle T(n,k) in which row n lists in increasing order the number of acyclic orientations of complete multipartite graphs K_lambda, where lambda is a partition of n into distinct parts; triangle T(n,k), n>=0, k = 1..A000009(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 8, 1, 16, 46, 1, 32, 146, 330, 1, 64, 454, 1066, 1374, 1, 128, 1394, 4718, 5658, 10554, 1, 256, 4246, 20266, 23118, 41506, 57054, 101502, 1, 512, 12866, 85310, 93930, 237686, 302730, 525642, 657210, 1165104, 1, 1024, 38854, 354106, 380094
Offset: 0

Views

Author

Alois P. Heinz, Apr 30 2024

Keywords

Comments

An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.

Examples

			Triangle T(n,k) begins:
  1;
  1;
  1;
  1,   4;
  1,   8;
  1,  16,   46;
  1,  32,  146,   330;
  1,  64,  454,  1066,  1374;
  1, 128, 1394,  4718,  5658, 10554;
  1, 256, 4246, 20266, 23118, 41506, 57054, 101502;
  ...

Links

Crossrefs

Columns k=1-2 give: A000012, A011782 (for n>=3).
Row sums give A370613.
Cf. A000009, A267383, A372254, A372261, A372326, A372396.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(
      expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
    end:
h:= proc() option remember; local q, l, b; q, l, b:= -1, args,
      proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*
        (q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))
      end; abs(b(0, nops(l)))
    end:
b:= proc(n, i, l) `if`(i*(i+1)/2 sort(b(n$2, [0]))[]:
seq(T(n), n=0..12);

A372395 Total number of acyclic orientations in all complete multipartite graphs K_lambda, where lambda ranges over all partitions of n.

Original entry on oeis.org

1, 1, 3, 11, 65, 411, 3535, 31081, 337185, 3846557, 50329253, 691740489, 10725769171, 172411994899, 3050277039465, 56428854605627, 1124781474310649, 23349607769846667, 518744693882444419, 11949343411110856153, 291921874093876965453, 7395266131906154621501
Offset: 0

Views

Author

Alois P. Heinz, Apr 29 2024

Keywords

Comments

An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.
All terms are odd.

Links

Crossrefs

Row sums of A372396.
Cf. A000041, A267383, A372254, A372261, A372326, A370613.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(
      expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
    end:
b:= proc(t, n, i) option remember; `if`(n=0, t!*(-1)^t,
     `if`(i<1, 0, b(t, n, i-1)+add(coeff(g(i), x, m)*
        b(t+m, n-i, min(n-i, i)), m=0..i)))
    end:
a:= n-> abs(b(0, n$2)):
seq(a(n), n=0..21);
  • Mathematica
    g[n_] := g[n] = If[n == 0, 1, Sum[Expand[x*g[n - j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
b[t_, n_, i_] := b[t, n, i] = If[n == 0, t!*(-1)^t, If[i < 1, 0, b[t, n, i - 1] + Sum[Coefficient[g[i], x, m]*b[t + m, n - i, Min[n - i, i]], {m, 0, i}]]];
a[n_] := Abs[b[0, n, n]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 24 2024, after Alois P. Heinz *)

A372396 Triangle T(n,k) in which row n lists in increasing order the number of acyclic orientations of complete multipartite graphs K_lambda, where lambda is a partition of n; triangle T(n,k), n>=0, k = 1..A000041(n), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 6, 1, 8, 14, 18, 24, 1, 16, 46, 54, 78, 96, 120, 1, 32, 146, 162, 230, 330, 384, 426, 504, 600, 720, 1, 64, 454, 486, 1066, 1374, 1536, 1902, 2286, 2616, 3000, 3216, 3720, 4320, 5040, 1, 128, 1394, 1458, 4718, 5658, 6144, 6902, 10554, 12090
Offset: 0

Views

Author

Alois P. Heinz, Apr 29 2024

Keywords

Comments

An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.

Examples

			Triangle T(n,k) begins:
  1;
  1;
  1,  2;
  1,  4,   6;
  1,  8,  14,  18,  24;
  1, 16,  46,  54,  78,  96, 120;
  1, 32, 146, 162, 230, 330, 384, 426, 504, 600, 720;
  ...

Links

Crossrefs

Columns k=1-3 give: A000012, A011782 (for n>=2), A027649(n-2) (for n>=4).
Row sums give A372395.
Cf. A000041, A000142, A001563, A267383, A372254, A372261, A372326, A370614.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(
      expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
    end:
h:= proc() option remember; local q, l, b; q, l, b:= -1, args,
      proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*
        (q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))
      end; abs(b(0, nops(l)))
    end:
b:= proc(n, i, l) `if`(n=0 or i=1, [h([l[], 1$n, 0])],
     [b(n-i, min(n-i, i), [l[], i])[], b(n, i-1, l)[]])
    end:
T:= n-> sort(b(n$2, []))[]:
seq(T(n), n=0..10);

Formula

T(n,A000041(n)) = A000142(n).
T(n,A000041(n)-1) = A001563(n-1) for n>=2.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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