login
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.
3
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
0,5
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
Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
EXAMPLE
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;
...
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<n, [], `if`(n=0, [h(l)],
[b(n-i, min(n-i, i-1), [l[], i])[], b(n, i-1, l)[]]))
end:
T:= n-> sort(b(n$2, [0]))[]:
seq(T(n), n=0..12);
CROSSREFS
Columns k=1-2 give: A000012, A011782 (for n>=3).
Row sums give A370613.
Sequence in context: A080102 A106475 A376302 * A134829 A245966 A130297
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Apr 30 2024
STATUS
approved