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

3,1

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

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.

Example

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;
  ...

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 *)

Crossrefs

Row sums give A038158.

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

Cf. A282507.

Sequence in context: A342786 A293474 A308727 * A254793 A353715 A182633

Adjacent sequences: A268214 A268215 A268216 * A268218 A268219 A268220

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 29 2016

Extensions

Title clarified by Geoffrey Critzer, Feb 19 2017

Corrected and extended by Andrew Howroyd, Aug 09 2025

Status

approved