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.

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

Author

Alois P. Heinz, Apr 24 2024

Keywords

• nonn
• look
• tabf

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

• Alois P. Heinz, Rows n = 0..40, flattened
• Don Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (19.2).
• Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
• Wikipedia, Acyclic orientation
• Wikipedia, Multipartite graph

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 *)