A268219 -id:A268219 - cp's OEIS frontend

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

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

A268218 a(n) = (n!/3!)*Sum(1/k!,k=1..n-3).

Original entry on oeis.org

0, 0, 0, 0, 4, 30, 200, 1435, 11536, 103908, 1039200, 11431365, 137176600, 1783296086, 24966145568, 374492183975, 5991874944160, 101861874051400, 1833513732926016, 34836760925595273, 696735218511906600, 14631439588750039930, 321891670952500880000, 7403508431907520241771

Offset: 0

N. J. A. Sloane, Jan 30 2016

nonn

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

For others in this series, see A038156, A038158, A268219, A268220.

g:=(r,n)->(n!/r!)*add(1/k!,k=1..n-r);
g2:=r->[seq(g(r,n),n=0..30)];
g2(3);

a(n) = (n!/3!)*sum(k=1, n-3, 1/k!); \\ Michel Marcus, Jan 30 2016

A268220 a(n) = (n!/5!)*Sum(1/k!,k=1..n-5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 63, 560, 5166, 51912, 571494, 6858720, 89164647, 1248307060, 18724608903, 299593746816, 5093093702060, 91675686645648, 1741838046278940, 34836760925594304, 731571979437500733, 16094583547625042460, 370175421595376010229, 8884210118289024288000

Offset: 0

N. J. A. Sloane, Jan 30 2016

nonn

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

For others in this series, see A038156, A038158, A268218, A268219.

g:=(r,n)->(n!/r!)*add(1/k!,k=1..n-r);
g2:=r->[seq(g(r,n),n=0..30)];
g2(5);

a(n) = (n!/5!)*sum(k=1, n-5, 1/k!); \\ Michel Marcus, Jan 30 2016

cp's OEIS Frontend

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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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

Extensions

A268218 a(n) = (n!/3!)*Sum(1/k!,k=1..n-3).

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A268220 a(n) = (n!/5!)*Sum(1/k!,k=1..n-5).

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Author

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Links

Crossrefs

Programs

Maple

PARI