A372396 - OEIS

%I #30 Apr 30 2024 18:44:25

%S 1,1,1,2,1,4,6,1,8,14,18,24,1,16,46,54,78,96,120,1,32,146,162,230,330,

%T 384,426,504,600,720,1,64,454,486,1066,1374,1536,1902,2286,2616,3000,

%U 3216,3720,4320,5040,1,128,1394,1458,4718,5658,6144,6902,10554,12090

%N 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; triangle T(n,k), n>=0, k = 1..A000041(n), read by rows.

%C 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.

%H Alois P. Heinz, <a href="/A372396/b372396.txt">Rows n = 0..28, flattened</a>

%H Richard P. Stanley, <a href="http://dx.doi.org/10.1016/0012-365X(73)90108-8">Acyclic Orientations of Graphs</a>, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Acyclic_orientation">Acyclic orientation</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multipartite_graph">Multipartite graph</a>

%F T(n,A000041(n)) = A000142(n).

%F T(n,A000041(n)-1) = A001563(n-1) for n>=2.

%e Triangle T(n,k) begins:

%e   1;

%e   1;

%e   1,  2;

%e   1,  4,   6;

%e   1,  8,  14,  18,  24;

%e   1, 16,  46,  54,  78,  96, 120;

%e   1, 32, 146, 162, 230, 330, 384, 426, 504, 600, 720;

%e   ...

%p g:= proc(n) option remember; `if`(n=0, 1, add(

%p       expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))

%p     end:

%p h:= proc() option remember; local q, l, b; q, l, b:= -1, args,

%p       proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*

%p         (q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))

%p       end; abs(b(0, nops(l)))

%p     end:

%p b:= proc(n, i, l) `if`(n=0 or i=1, [h([l[], 1$n, 0])],

%p      [b(n-i, min(n-i, i), [l[], i])[], b(n, i-1, l)[]])

%p     end:

%p T:= n-> sort(b(n$2, []))[]:

%p seq(T(n), n=0..10);

%Y Columns k=1-3 give: A000012, A011782 (for n>=2), A027649(n-2) (for n>=4).

%Y Row sums give A372395.

%Y Cf. A000041, A000142, A001563, A267383, A372254, A372261, A372326, A370614.

%K nonn,tabf

%O 0,4

%A _Alois P. Heinz_, Apr 29 2024