A281779 Number of distinct topologies on an n-set that have exactly 11 open sets.

0, 0, 0, 0, 0, 500, 16980, 342160, 5486040, 77926380, 1031160060, 13047426920, 160124426880, 1921105846660, 22632779709540, 262513678889280, 3002768326532520, 33914184260797340, 378596540805849420, 4181330954328313240, 45727913513193402960, 495618273676457274420

Offset: 0

Author

Geoffrey Critzer, Jan 29 2017

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..950
• Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
• Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Crossrefs

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.

Programs

PARI

a(n) = 25*5!*stirling(n, 5, 2)/6 + 79*6!*stirling(n, 6, 2)/6 + 29*7!*stirling(n, 7, 2)/2 + 39*8!*stirling(n, 8, 2)/4 + 4*9!*stirling(n, 9, 2) + 10!*stirling(n, 10, 2) \\ Colin Barker, Jan 30 2017

PARI

concat(vector(4), Vec(20*x^5*(25 - 526*x + 3413*x^2 + 292*x^3 - 72756*x^4 + 226800*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)*(1 - 10*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = 25/6*5! Stirling2(n, 5) + 79/6*6! Stirling2(n, 6) + 29/2*7! Stirling2(n, 7) + 39/4*8! Stirling2(n, 8) + 4*9! Stirling2(n, 9) + 10! Stirling2(n, 10).

G.f.: 20*x^5*(25 - 526*x + 3413*x^2 + 292*x^3 - 72756*x^4 + 226800*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)*(1 - 10*x)). - Colin Barker, Jan 30 2017