OFFSET
0,3
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.
LINKS
Alois P. Heinz, Rows n = 0..140, 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
EXAMPLE
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, ...
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 24 2024
STATUS
approved