A281778 Number of distinct topologies on an n-set that have exactly 10 open sets.

0, 0, 0, 0, 24, 900, 18030, 276570, 3680964, 45065160, 523292010, 5859909990, 63862084704, 680829769620, 7122705252390, 73284607133010, 742843170653244, 7429450873589280, 73416173732059170, 717721593866613630, 6949589106333898584, 66721599431782204140

Offset: 0

Author

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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 (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).

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

concat(vector(4), Vec(6*x^4*(4 - 30*x - 265*x^2 + 3570*x^3 - 10839*x^4 + 22680*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)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = 4! Stirling2(n, 4) + 11/2*5! Stirling2(n, 5) + 73/8*6! Stirling2(n, 6) + 15/2*7! Stirling2(n, 7) + 7/2*8! Stirling2(n, 8) + 9! Stirling2(n, 9).

G.f.: (6*(4 - 30*x - 265*x^2 + 3570*x^3 - 10839*x^4 + 22680*x^5))*x^4/Product_{j=1..9} (1-j*x). - Robert Israel, Jan 29 2017