A268217 -id:A268217 - cp's OEIS frontend

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

Lekraj Beedassy, Jul 29 2006

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]

			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;
  ...

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.

Row sums give A038156.
Triangles in this series: A268216, A268217, A268221, A268222, A268223.
Cf. A008279, A014631, A282507.

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

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

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

Edited by Don Reble, Aug 01 2006

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.

Original entry on oeis.org

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

N. J. A. Sloane, Jan 30 2016

			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;
  ...

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.

Row sums give A268218.
Triangles in this series: A119741, A268217, A268221, A268222, A268223.
Cf. A282507.

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 *)

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

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.

Original entry on oeis.org

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

N. J. A. Sloane, Jan 30 2016

			Triangle begins:
    5;
   15,   30;
   35,  105,  210;
   70,  280,  840,  1680;
  126,  630, 2520,  7560, 15120;
  210, 1260, 6300, 25200, 75600, 151200;
...

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.

Row sums give A268219.
Triangles in this series: A119741, A268217, A268221, A268222, A268223.
Cf. A282507.

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 *)

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

A268223 Triangle read by rows: T(n,k) (n>=6, k=3..n-3) 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 {0,5,6,7,...,s,n} where s is the size of the largest proper open set in t.

Original entry on oeis.org

6, 21, 42, 56, 168, 336, 126, 504, 1512, 3024, 252, 1260, 5040, 15120, 30240, 462, 2772, 13860, 55440, 166320, 332640, 792, 5544, 33264, 166320, 665280, 1995840, 3991680, 1287, 10296, 72072, 432432, 2162160, 8648640, 25945920, 51891840

Offset: 6

N. J. A. Sloane, Jan 30 2016

			Triangle begins:
    6;
   21,   42;
   56,  168,  336;
  126,  504, 1512,  3024;
  252, 1260, 5040, 15120, 30240;
  ...

Andrew Howroyd, Table of n, a(n) for n = 6..1280 (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.

Row sums give A268219.

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

i = 5; Table[ Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0,
n - i - 1}], {n, 2, 13}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)

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

A038158 a(n) = (n!/2)*Sum(1/k!, k=1..n-2).

Original entry on oeis.org

0, 0, 0, 3, 18, 100, 615, 4326, 34636, 311760, 3117645, 34294150, 411529866, 5349888336, 74898436795, 1123476552030, 17975624832600, 305585622154336, 5500541198778201, 104510282776785990, 2090205655535719990, 43894318766250120000

Offset: 0

N. J. A. Sloane

nonn

G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Row sums of A268217.

Cf. A038157.

a(n) = (n!/2)*sum(k=1, n-2, 1/k!); \\ Michel Marcus, Jan 30 2016

a(n) = A038157(n) / 2. - Sean A. Irvine, Jan 09 2021

cp's OEIS Frontend

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

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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.

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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.

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Mathematica

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A268223 Triangle read by rows: T(n,k) (n>=6, k=3..n-3) 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 {0,5,6,7,...,s,n} where s is the size of the largest proper open set in t.

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Programs

Mathematica

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A038158 a(n) = (n!/2)*Sum(1/k!, k=1..n-2).

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