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

6,1

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.

Example

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

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 *)

Crossrefs

Row sums give A268219.

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

Sequence in context: A364169 A364171 A199194 * A048036 A272671 A272684

Adjacent sequences: A268220 A268221 A268222 * A268224 A268225 A268226

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 30 2016

Extensions

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

Status

approved