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, 0, 1, 62, 571, 2100, 4680, 7830, 11760, 14260, 18030, 16980

Offset

0,6

Links

Andrew Howroyd, Table of n, a(n) for n = 0..254

Wikipedia, Topological space

Formula

T(n, k) = # { t in A101620(n,.) | A000120(t)==k+1 }, where A101620(n,.) = {A101620(n,j); j = 1,...,A000798(n)} is the row of that table and A000120 is the Hammingweight a.k.a. BitCount function. - M. F. Hasler, Jun 21 2026

Example

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}

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}]

Prog

(PARI) A326882(n, k)=sum(i=1, #n=A101620_row(n), hammingweight(n[i])==k+1)
A326882_row(n)={my(c=Vec(0, 2^n)); foreach(A101620_row(n), t, c[hammingweight(t)]++); c} \\ M. F. Hasler, Jun 21 2026
(Python)
A326882=lambda n, k: sum(t.bit_count()==k+1 for t in A101620_row(n)) # M. F. Hasler, Jun 21 2026
def A326882_row(n):
    c = [0]*2**n
    for t in A101620_row(n): c[t.bit_count()-1] += 1
    return c # M. F. Hasler, Jun 21 2026

Crossrefs

Row lengths are A000079.

Row sums are A000798.

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

Columns 0 - 11 are: A000007, A057427, A000918 (n>0), A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

Sequence in context: A011126 A395416 A266854 * A265170 A217653 A392515

Adjacent sequences: A326879 A326880 A326881 * A326883 A326884 A326885

Keyword

nonn,tabf,nice,changed

Author

Gus Wiseman, Aug 01 2019

Extensions

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

Status

approved