A268222 Triangle read by rows: T(n,k) (n>=5, k=3..n-2) 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 {0,4,5,6,...,s,n} where s is the size of the largest proper open set in t.

5, 15, 30, 35, 105, 210, 70, 280, 840, 1680, 126, 630, 2520, 7560, 15120, 210, 1260, 6300, 25200, 75600, 151200, 330, 2310, 13860, 69300, 277200, 831600, 1663200, 495, 3960, 27720, 166320, 831600, 3326400, 9979200, 19958400, 715, 6435, 51480, 360360, 2162160, 10810800, 43243200, 129729600, 259459200

Offset

5,1

Links

Andrew Howroyd, Table of n, a(n) for n = 5..1279 (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:
    5;
   15,   30;
   35,  105,  210;
   70,  280,  840,  1680;
  126,  630, 2520,  7560, 15120;
  210, 1260, 6300, 25200, 75600, 151200;
...

Mathematica

i = 4; 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 A268219.

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

Cf. A282507.

Sequence in context: A162525 A212871 A188350 * A285630 A078905 A059160

Adjacent sequences: A268219 A268220 A268221 * A268223 A268224 A268225

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 30 2016

Extensions

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

Missing a(19) inserted and a(41) onwards from Andrew Howroyd, Aug 10 2025

Status

approved