A266858 -id:A266858 - cp's OEIS frontend

A267383 Number A(n,k) of acyclic orientations of the Turán graph T(n,k); square array A(n,k), n>=0, k>=1, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 6, 14, 1, 1, 1, 2, 6, 18, 46, 1, 1, 1, 2, 6, 24, 78, 230, 1, 1, 1, 2, 6, 24, 96, 426, 1066, 1, 1, 1, 2, 6, 24, 120, 504, 2286, 6902, 1, 1, 1, 2, 6, 24, 120, 600, 3216, 15402, 41506, 1

Offset: 0

Alois P. Heinz, Jan 13 2016

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.

Conjecture: In general, column k > 1 is asymptotic to n! / ((k-1) * (1 - log(k/(k-1)))^((k-1)/2) * k^n * (log(k/(k-1)))^(n+1)). - Vaclav Kotesovec, Feb 18 2017

			Square array A(n,k) begins:
  1,    1,    1,    1,    1,    1,    1, ...
  1,    1,    1,    1,    1,    1,    1, ...
  1,    2,    2,    2,    2,    2,    2, ...
  1,    4,    6,    6,    6,    6,    6, ...
  1,   14,   18,   24,   24,   24,   24, ...
  1,   46,   78,   96,  120,  120,  120, ...
  1,  230,  426,  504,  600,  720,  720, ...
  1, 1066, 2286, 3216, 3720, 4320, 5040, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
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, Turán graph

Columns k=1-10 give: A000012, A266695, A266858, A267384, A267385, A267386, A267387, A267388, A267389, A267390.
Main diagonal gives A000142.
A(2n,n) gives A033815.
A(n,ceiling(n/2)) gives A161132.
Bisection of column k=2 gives A048163.
Trisection of column k=3 gives A370961.
a(n^2,n) gives A372084.

A:= proc(n, k) option remember; local b, l, q; q:=-1;
       l:= [floor(n/k)$(k-irem(n,k)), ceil(n/k)$irem(n,k)];
       b:= proc(n, j) option remember; `if`(j=1, (q-n)^l[1]*
             mul(q-i, i=0..n-1), add(b(n+m, j-1)*
             Stirling2(l[j], m), m=0..l[j]))
           end; forget(b);
       abs(b(0, k))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = Module[{ b, l, q}, q = -1; l = Join[Array[Floor[n/k] &, k - Mod[n, k]], Array[ Ceiling[n/k] &, Mod[n, k]]]; b[nn_, j_] := b[nn, j] = If[j == 1, (q - nn)^l[[1]]*Product[q - i, {i, 0, nn - 1}], Sum[b[nn + m, j - 1]*StirlingS2[l[[j]], m], {m, 0, l[[j]]}]]; Abs[b[0, k]]]; Table[Table[A[n, 1 + d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A266695 Number of acyclic orientations of the Turán graph T(n,2).

Original entry on oeis.org

1, 1, 2, 4, 14, 46, 230, 1066, 6902, 41506, 329462, 2441314, 22934774, 202229266, 2193664790, 22447207906, 276054834902, 3216941445106, 44222780245622, 578333776748674, 8787513806478134, 127464417117501586, 2121181056663291350, 33800841048945424546

Offset: 0

Alois P. Heinz, Jan 02 2016

nonn

The Turán graph T(n,2) is also the complete bipartite graph K_{floor(n/2),ceiling(n/2)}.

Alois P. Heinz, Table of n, a(n) for n = 0..475
Beáta Bényi and Peter Hajnal, Combinatorics of poly-Bernoulli numbers, arXiv:1510.05765 [math.CO], 2015; Studia Scientiarum Mathematicarum Hungarica, Vol. 52, No. 4 (2015), 537-558, DOI:10.1556/012.2015.52.4.1325.
P. J. Cameron, C. A. Glass, and R. U. Schumacher, Acyclic orientations and poly-Bernoulli numbers, arXiv:1412.3685 [math.CO], 2014-2018.
Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Wikipedia, Turán graph

Column k=2 of A267383.
Cf. A099594, A212084, A212085, A266858, A266972.
Bisections give A048163 (even part), A188634 (odd part).

a:= n-> (p-> add(Stirling2(n-p+1, i+1)*Stirling2(p+1, i+1)*
         i!^2, i=0..p))(iquo(n, 2)):
seq(a(n), n=0..25);

a[n_] := With[{q=Quotient[n, 2]}, Sum[StirlingS2[n-q+1, i+1] StirlingS2[ q+1, i+1] i!^2, {i, 0, q}]];
Array[a, 24, 0] (* Jean-François Alcover, Nov 06 2018 *)

a(n) = Sum_{i=0..floor(n/2)} i!^2 * Stirling2(ceiling(n/2)+1,i+1) * Stirling2(floor(n/2)+1,i+1).
a(n) = A099594(floor(n/2),ceiling(n/2)).
a(n) = Sum_{k=0..n} abs(A266972(n,k)).
a(n) ~ n! / (sqrt(1-log(2)) * 2^n * (log(2))^(n+1)). - Vaclav Kotesovec, Feb 18 2017

A370961 a(n) = number of acyclic orientations of the complete tripartite graph K_{n,n,n}.

Original entry on oeis.org

1, 6, 426, 122190, 90768378, 138779942046, 379578822373866, 1689637343582548590, 11434884125767376107098, 111765072808554847704145086, 1515592947854931941485836600906, 27609710924806869786487193747541390, 658043992934027491354757341987635993018

Offset: 0

N. J. A. Sloane, Apr 04 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..155 (terms n = 1..16 from Don Knuth)
Don Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (19.2).

Cf. A212220, A266858, A267383, A370960.
Main diagonal of A372254.
Row n=3 of A372326.

a(n) = A266858(3n) = A267383(3n,3). - Alois P. Heinz, Apr 17 2024

a(n) = Sum_{k=0..3n} (-1)^k * A212220(n,k). - Alois P. Heinz, May 02 2024

More terms from Don Knuth, Apr 07 2024

a(0)=1 prepended by Alois P. Heinz, Apr 17 2024

cp's OEIS Frontend

A267383 Number A(n,k) of acyclic orientations of the Turán graph T(n,k); square array A(n,k), n>=0, k>=1, read by antidiagonals.

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A266695 Number of acyclic orientations of the Turán graph T(n,2).

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Maple

Mathematica

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A370961 a(n) = number of acyclic orientations of the complete tripartite graph K_{n,n,n}.

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