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A326882 Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.

%I A326882 #20 Feb 26 2020 13:38:30
%S A326882 1,0,1,0,1,2,1,0,1,6,9,6,6,0,1,0,1,14,43,60,72,54,54,20,24,0,12,0,0,0,
%T A326882 1,0,1,30,165,390,630,780,955,800,900,500,660,240,390,120,190,10,100,
%U A326882 0,60,0,0,0,20,0,0,0,0,0,0,0,1
%N A326882 Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.
%H A326882 Andrew Howroyd, <a href="/A326882/b326882.txt">Table of n, a(n) for n = 0..254</a>
%H A326882 Wikipedia, <a href="https://en.wikipedia.org/wiki/Topological_space">Topological space</a>
%e A326882 Triangle begins:
%e A326882   1
%e A326882   0  1
%e A326882   0  1  2  1
%e A326882   0  1  6  9  6  6  0  1
%e A326882   0  1 14 43 60 72 54 54 20 24  0 12  0  0  0  1
%e A326882 Row n = 3 counts the following topologies:
%e A326882 {}{123} {}{1}{123}  {}{1}{12}{123} {}{1}{2}{12}{123}  {}{1}{2}{12}{13}{123}
%e A326882         {}{2}{123}  {}{1}{13}{123} {}{1}{3}{13}{123}  {}{1}{2}{12}{23}{123}
%e A326882         {}{3}{123}  {}{1}{23}{123} {}{2}{3}{23}{123}  {}{1}{3}{12}{13}{123}
%e A326882         {}{12}{123} {}{2}{12}{123} {}{1}{12}{13}{123} {}{1}{3}{13}{23}{123}
%e A326882         {}{13}{123} {}{2}{13}{123} {}{2}{12}{23}{123} {}{2}{3}{12}{23}{123}
%e A326882         {}{23}{123} {}{2}{23}{123} {}{3}{13}{23}{123} {}{2}{3}{13}{23}{123}
%e A326882                     {}{3}{12}{123}
%e A326882                     {}{3}{13}{123}      {}{1}{2}{3}{12}{13}{23}{123}
%e A326882                     {}{3}{23}{123}
%t A326882 Table[Length[Select[Subsets[Subsets[Range[n]],{k}],MemberQ[#,{}]&&MemberQ[#,Range[n]]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4},{k,2^n}]
%Y A326882 Row lengths are A000079.
%Y A326882 Row sums are A000798.
%Y A326882 Cf. A001930, A014466, A102894, A102895, A102896, A102897, A306445, A326876, A326878, A326881.
%Y A326882 Columns: A281774 and refs therein.
%K A326882 nonn,tabf,nice
%O A326882 0,6
%A A326882 _Gus Wiseman_, Aug 01 2019
%E A326882 Terms a(31) and beyond from _Andrew Howroyd_, Aug 10 2019