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.

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, 1, 0, 1, 30, 165, 390, 630, 780, 955, 800, 900, 500, 660, 240, 390, 120, 190, 10, 100, 0, 60, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Gus Wiseman, Aug 01 2019

Keywords

• nonn
• tabf
• nice

Examples

			Triangle begins:
  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  1
Row n = 3 counts the following topologies:
{}{123} {}{1}{123}  {}{1}{12}{123} {}{1}{2}{12}{123}  {}{1}{2}{12}{13}{123}
        {}{2}{123}  {}{1}{13}{123} {}{1}{3}{13}{123}  {}{1}{2}{12}{23}{123}
        {}{3}{123}  {}{1}{23}{123} {}{2}{3}{23}{123}  {}{1}{3}{12}{13}{123}
        {}{12}{123} {}{2}{12}{123} {}{1}{12}{13}{123} {}{1}{3}{13}{23}{123}
        {}{13}{123} {}{2}{13}{123} {}{2}{12}{23}{123} {}{2}{3}{12}{23}{123}
        {}{23}{123} {}{2}{23}{123} {}{3}{13}{23}{123} {}{2}{3}{13}{23}{123}
                    {}{3}{12}{123}
                    {}{3}{13}{123}      {}{1}{2}{3}{12}{13}{23}{123}
                    {}{3}{23}{123}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..254
• Wikipedia, Topological space

Crossrefs

Row lengths are A000079.

Row sums are A000798.

Cf. A001930, A014466, A102894, A102895, A102896, A102897, A306445, A326876, A326878, A326881.

Columns: A281774 and refs therein.

Programs

Mathematica

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}]

Extensions

Terms a(31) and beyond from Andrew Howroyd, Aug 10 2019