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A370613 Total number of acyclic orientations in all complete multipartite graphs K_lambda, where lambda ranges over all partitions of n into distinct parts. 3
1, 1, 1, 5, 9, 63, 509, 2959, 22453, 247949, 3080991, 28988331, 407320739, 5122243495, 82583577967, 1430027615585, 22556817627789, 395098668828675, 7979894546677853, 154786744386253387, 3355612019167352821, 78865333300205585345, 1769663675666499515751 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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
Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
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<n, 0,
`if`(n=0, t!*(-1)^t, add(coeff(g(i), x, m)*
b(t+m, n-i, min(n-i, i-1)), m=0..i)+b(t, n, i-1)))
end:
a:= n-> abs(b(0, n$2)):
seq(a(n), n=0..22);
CROSSREFS
Row sums of A370614.
Sequence in context: A222536 A222698 A200440 * A347991 A179100 A358966
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 30 2024
STATUS
approved

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Last modified June 18 04:26 EDT 2024. Contains 373468 sequences. (Running on oeis4.)