A268217 Triangle read by rows: T(n,k) (n>=3, k=3..n) 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 {0,2,3,4,...,s,n} where s is the size of the largest proper open set in t.

3, 6, 12, 10, 30, 60, 15, 60, 180, 360, 21, 105, 420, 1260, 2520, 28, 168, 840, 3360, 10080, 20160, 36, 252, 1512, 7560, 30240, 90720, 181440, 45, 360, 2520, 15120, 75600, 302400, 907200, 1814400, 55, 495, 3960, 27720, 166320, 831600, 3326400, 9979200, 19958400

Offset: 3

Author

N. J. A. Sloane, Jan 29 2016

Keywords

• nonn
• tabl
• changed

Comments

When two leading 0's are added and last element repeated, rows give the coefficients of the path polynomials of the complete graph K_n. - Eric W. Weisstein, Jun 04 2017

Examples

			Triangle begins:
   3;
   6,  12;
  10,  30,   60;
  15,  60,  180,   360;
  21, 105,  420,  1260,   2520;
  28, 168,  840,  3360,  10080,  20160;
  36, 252, 1512,  7560,  30240,  90720,  181440;
  45, 360, 2520, 15120,  75600, 302400,  907200, 1814400;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..1277 (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.
• Eric Weisstein's World of Mathematics, Complete Graph.
• Eric Weisstein's World of Mathematics, Graph Path.

Crossrefs

Row sums give A038158.

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

Cf. A282507.

Programs

Mathematica

i = 2; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 9}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)
CoefficientList[Table[-(1/2) (n - 1) n x^(n - 2) (Gamma[n - 1] - E^(1/x) Gamma[n - 1, 1/x]), {n, 3, 10}] // FunctionExpand, x] // Flatten (* Eric W. Weisstein, Jun 04 2017 *)

Extensions

Title clarified by Geoffrey Critzer, Feb 19 2017

Corrected and extended by Andrew Howroyd, Aug 09 2025