A119741 A008279, with the first and last of each row removed.

2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 6, 30, 120, 360, 720, 7, 42, 210, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800

Offset

2,1

Comments

Triangle read by rows: T(n,k) (n>=2, k=1..n-1) is the number of topologies t on n points having exactly k+2 open sets such that t contains exactly one open set of size m for each m in {0,1,2,...,s,n} where s is the size of the largest proper open set in t. - N. J. A. Sloane, Jan 29 2016 [clarified by Geoffrey Critzer, Feb 19 2017]

Links

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (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.

Formula

a(n) = (A003057(n))!/(A004736(n))! = (A002260(n))!*(A014410(n)).

T(n,k) = A173333(n+1,n-k+1), 1<=k<=n. - Reinhard Zumkeller, Feb 19 2010

Example

Triangle begins:
   2;
   3,  6;
   4, 12,  24;
   5, 20,  60,  120;
   6, 30, 120,  360,   720;
   7, 42, 210,  840,  2520,   5040;
   8, 56, 336, 1680,  6720,  20160,  40320;
   9, 72, 504, 3024, 15120,  60480, 181440,  362880;
  10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800;
  ...

Maple

T:= (n, k)-> n!/(n-k)!:
seq(seq(T(n, k), k=1..n-1), n=2..11);  # Alois P. Heinz, Aug 22 2025

Mathematica

Table[FactorialPower[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 21 2020 *)

Crossrefs

Row sums give A038156.

Triangles in this series: A268216, A268217, A268221, A268222, A268223.

Cf. A008279, A014631, A282507.

Sequence in context: A097275 A130879 A268216 * A346928 A375761 A245712

Adjacent sequences: A119738 A119739 A119740 * A119742 A119743 A119744

Keyword

nonn,tabl

Author

Lekraj Beedassy, Jul 29 2006

Extensions

Edited by Don Reble, Aug 01 2006

Status

approved