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 1-3 of 3 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

A372254 Number A(n,k) of acyclic orientations of the complete tripartite graph K_{n,n,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 6, 14, 1, 18, 78, 230, 1, 54, 426, 1902, 6902, 1, 162, 2286, 15402, 76110, 329462, 1, 486, 12090, 122190, 822954, 4553166, 22934774, 1, 1458, 63198, 951546, 8724078, 61796298, 381523758, 2193664790, 1, 4374, 327306, 7290942, 90768378, 823457454, 6241779786, 42700751022, 276054834902
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

			Square array A(n,k) begins:
       1,       1,        1,         1,           1,            1, ...
       2,       6,       18,        54,         162,          486, ...
      14,      78,      426,      2286,       12090,        63198, ...
     230,    1902,    15402,    122190,      951546,      7290942, ...
    6902,   76110,   822954,   8724078,    90768378,    928340190, ...
  329462, 4553166, 61796298, 823457454, 10779805722, 138779942046, ...

Links

Crossrefs

Rows n=0-2 give: A000012, A008776, A370960.
Column k=0 gives A048163(n+1).
Main diagonal gives A370961.
Cf. A212220, A267383, A372261.

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:
A:= proc(n, k) option remember; local q, l, b; q, l, b:= -1, [n$2, k],
      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, 3))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..9);
  • Mathematica
    g[n_] := g[n] = If[n == 0, 1, Sum[Expand[x*g[n-j]]*Binomial[n-1, j-1], {j, 1, n}]];
A[n_, k_] := A[n, k] = Module[{q, l, b}, {q, l} = {-1, {n, n, k}}; b[n0_, j_] := b[n0, j] = If[j == 1, Product[q-i, {i, 0, n0-1}]*(q-n0)^l[[1]], Sum[b[n0 + m, j-1]*Coefficient[g[l[[j]]], x, m], {m, 0, l[[j]]}]]; Abs[b[0, 3]]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 9}] // Flatten (* Jean-François Alcover, Apr 25 2024, after Alois P. Heinz *)

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 *)
Showing 1-3 of 3 results.

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

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