A268223 Triangle read by rows: T(n,k) (n>=6, k=3..n-3) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,5,6,7,...,s,n} where s is the size of the largest proper open set in t.

6, 21, 42, 56, 168, 336, 126, 504, 1512, 3024, 252, 1260, 5040, 15120, 30240, 462, 2772, 13860, 55440, 166320, 332640, 792, 5544, 33264, 166320, 665280, 1995840, 3991680, 1287, 10296, 72072, 432432, 2162160, 8648640, 25945920, 51891840

Offset: 6

Author

N. J. A. Sloane, Jan 30 2016

Keywords

• nonn
• tabl
• changed

Examples

			Triangle begins:
    6;
   21,   42;
   56,  168,  336;
  126,  504, 1512,  3024;
  252, 1260, 5040, 15120, 30240;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 6..1280 (first 50 rows)
• G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Crossrefs

Row sums give A268219.

Triangles in this series: A119741, A268217, A268221, A268222, A268223.

Programs

Mathematica

i = 5; Table[ Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0,
n - i - 1}], {n, 2, 13}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)

Extensions

Title clarified and more terms added by Geoffrey Critzer, Feb 19 2017