A268221 Triangle read by rows: T(n,k) (n>=4, k=3..n-1) 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 {0,3,4,5,...,s,n} where s is the size of the largest proper open set in t.

4, 10, 20, 20, 60, 120, 35, 140, 420, 840, 56, 280, 1120, 3360, 6720, 84, 504, 2520, 10080, 30240, 60480, 120, 840, 5040, 25200, 100800, 302400, 604800, 165, 1320, 9240, 55440, 277200, 1108800, 3326400, 6652800, 220, 1980, 15840, 110880, 665280, 3326400, 13305600, 39916800, 79833600

Offset

4,1

Links

Andrew Howroyd, Table of n, a(n) for n = 4..1278 (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:
    4;
   10,  20;
   20,  60,  120;
   35, 140,  420,   840;
   56, 280, 1120,  3360,   6720;
   84, 504, 2520, 10080,  30240,  60480;
  120, 840, 5040, 25200, 100800, 302400, 604800;
  ...

Mathematica

i = 3; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 12}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)

Crossrefs

Row sums give A268218.

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

Cf. A282507.

Sequence in context: A155229 A155220 A300018 * A086176 A015789 A145021

Adjacent sequences: A268218 A268219 A268220 * A268222 A268223 A268224

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