A102897 -id:A102897 - cp's OEIS frontend

A108800 Number of nonisomorphic systems enumerated by A102895.

1, 2, 6, 28, 330, 28960, 216562364, 5592326182940100

Offset: 0

Don Knuth, Jul 01 2005

Also the number of non-isomorphic sets of sets with {} that are closed under intersection. Also the number of non-isomorphic set-systems (without {}) covering n + 1 vertices and closed under intersection. - Gus Wiseman, Aug 05 2019

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 28 sets of sets with {} that are closed under intersection:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{1}{2}      {}{123}
             {}{2}{12}     {}{1}{2}
             {}{1}{2}{12}  {}{1}{23}
                           {}{2}{12}
                           {}{3}{123}
                           {}{1}{2}{3}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{1}{3}{23}
                           {}{2}{3}{123}
                           {}{3}{13}{23}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{1}{2}{3}{23}
                           {}{1}{2}{3}{123}
                           {}{2}{3}{13}{23}
                           {}{1}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{1}{2}{3}{13}{23}
                           {}{1}{2}{3}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}
                           {}{1}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

Except a(0) = 1, first differences of A193675.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The labeled version is A102895.
The case also closed under union is A326898.
The covering case is A326883.
Cf. A001930, A102894, A102896, A102897, A193674, A326880, A326881.

a(n > 0) = 2 * A108798(n).

a(6) added (using A193675) by N. J. A. Sloane, Aug 02 2011
Changed a(0) from 2 to 1 by Gus Wiseman, Aug 02 2019
a(7) added (using A108798) by Andrew Howroyd, Aug 10 2019

A326875 BII-numbers of set-systems that are closed under union.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 24, 25, 32, 34, 40, 42, 64, 65, 66, 68, 69, 70, 71, 72, 76, 80, 81, 82, 84, 85, 86, 87, 88, 89, 92, 93, 96, 97, 98, 100, 101, 102, 103, 104, 106, 108, 110, 112, 113, 114, 116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128

Offset: 1

Gus Wiseman, Jul 29 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

The enumeration of these set-systems by number of covered vertices is A102896.

			The sequence of all set-systems that are closed under union together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
  72: {{3},{1,2,3}}
  76: {{1,2},{3},{1,2,3}}
  80: {{1,3},{1,2,3}}
  81: {{1},{1,3},{1,2,3}}
  82: {{2},{1,3},{1,2,3}}
  84: {{1,2},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  86: {{2},{1,2},{1,3},{1,2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..3000

Cf. A006126, A048793, A102894, A102896, A102897, A326031, A326872, A326874, A326876, A326880.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union@@@Tuples[bpe/@bpe[#],2]]&]

from itertools import count, islice, combinations
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        E,f = [bin_i(k) for k in bin_i(n)],0
        for i in combinations(E,2):
            if list(set(i[0])|set(i[1])) not in E:
                f += 1
                break
        if f < 1:
            yield n
A326875_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Mar 06 2025

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.

Original entry on oeis.org

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

Gus Wiseman, Aug 01 2019

			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}

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

Row lengths are A000079.
Row sums are A000798.
Cf. A001930, A014466, A102894, A102895, A102896, A102897, A306445, A326876, A326878, A326881.
Columns: A281774 and refs therein.

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}]

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

A326880 BII-numbers of set-systems that are closed under nonempty intersection.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 46, 47, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88

Offset: 1

Gus Wiseman, Jul 29 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

The enumeration of these set-systems by number of covered vertices is A326881.

			Most small numbers are in the sequence, but the sequence of non-terms together with the set-systems with those BII-numbers begins:
  20: {{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  28: {{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  48: {{1,3},{2,3}}
  49: {{1},{1,3},{2,3}}
  50: {{2},{1,3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  84: {{1,2},{1,3},{1,2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A006126, A048793, A102894, A102895, A102896, A102897, A306445, A326031, A326872, A326874, A326875, A326876, A326881.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Intersection@@@Select[Tuples[bpe/@bpe[#],2],Intersection@@#!={}&]]&]

from itertools import count, islice, combinations
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        E,f = [bin_i(k) for k in bin_i(n)],0
        for i in combinations(E,2):
            x = list(set(i[0])&set(i[1]))
            if x not in E and len(x) > 0:
                f += 1
                break
        if f < 1:
            yield n
A326880_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Mar 07 2025

A326881 Number of set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 5, 71, 4223, 2725521, 151914530499, 28175294344381108057

Offset: 0

Gus Wiseman, Jul 30 2019

			The a(2) = 5 set-systems:
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{},{1},{2},{1,2}}

The case also closed under union is A000798.
The connected case (i.e., with maximum) is A102894.
The same for union instead of intersection is (also) A102894.
The non-covering case is A102895.
The BII-numbers of these set-systems (without the empty set) are A326880.
The unlabeled case is A326883.
Cf. A003465, A014466, A102896, A102897, A193674, A193675, A306445, A307249, A326878.

Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A102895. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A326906 Number of sets of subsets of {1..n} that are closed under union and cover all n vertices.

Original entry on oeis.org

2, 2, 8, 90, 4542, 2747402, 151930948472, 28175295407840207894

Offset: 0

Gus Wiseman, Aug 03 2019

Differs from A102895 in having a(0) = 2 instead of 1.

			The a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1,2}}
  {{}}  {{},{1}}  {{},{1,2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The case without empty sets is A102894.
The case with a single covering edge is A102895.
Binomial transform is A102897.
The case also closed under intersection is A326878 for n > 0.
The same for intersection instead of union is (also) A326906.
The unlabeled version is A326907.
Cf. A000798, A102896, A102897, A108800, A193675, A306445, A326880, A326881, A326883.

Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = 2 * A102894(n).

A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 4, 22, 302, 28630, 216533404, 5592325966377736

Offset: 0

Gus Wiseman, Jul 30 2019

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 22 set-systems:
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{}{1}{2}}      {{}{1}{23}}
                 {{}{2}{12}}     {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{1}{2}{3}}
                                 {{}{23}{123}}
                                 {{}{1}{3}{23}}
                                 {{}{2}{3}{123}}
                                 {{}{3}{13}{23}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{1}{2}{3}{23}}
                                 {{}{1}{2}{3}{123}}
                                 {{}{2}{3}{13}{23}}
                                 {{}{1}{3}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{13}{23}}
                                 {{}{1}{2}{3}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}}
                                 {{}{1}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The case also closed under union is A001930.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The non-covering case is A108800.
The labeled case is A326881.
Cf. A000798, A102894, A102895, A102897, A193674, A193675, A306445, A326875, A326878, A326880.

a(n) = A108800(n) - A108800(n-1) for n > 0. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

Original entry on oeis.org

2, 4, 12, 96, 4404, 2725942, 151906396568, 28175293281055562650

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A102897.
The version not closed under intersection is A326941.
The covering case is A326943.
The case without empty edges is A326959.
Cf. A003180, A182507, A316978, A319564, A326906, A326939, A326940, A326944, A326947.

Table[Length[Select[Subsets[Subsets[Range[n]]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Binomial transform of A326943.

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A109457 Number of Krom functions on n variables (or 2SAT instances): conjunctions of clauses with two literals per clause.

Original entry on oeis.org

2, 4, 16, 166, 4170, 224716, 24445368, 5167757614, 2061662323954

Offset: 0

Don Knuth, Aug 24 2005

A Krom function is equivalent to a Boolean function with the property that, if f(x)=f(y)=f(z)=1, then f()=1, where denotes the bitwise median of the three Boolean vectors x, y, z.

Also related to number of retracts of an n-cube (see Feder).

Tomas Feder, Stable Networks and Product Graphs, Memoirs of the American Mathematical Society, 555 (1995), Section 3.2.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
Knuth, Donald E., Satisfiability, Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, pages 148 and 220, Problem 191.
M. R. Krom, The decision problem for a class of first-order formulas in which all disjunctions are binary, Zeitschrift f. mathematische Logik und Grundlagen der Mathematik, 13 (1967), 15-20.
Thomas J. Schaefer, The complexity of satisfiability problems, ACM Symposium on Theory of Computing, 10 (1978), 216-226.

Cf. A109458, A109459, A102897.

Cf. A112535.

A102501 Primes of the concatenated form 7nn7.

Original entry on oeis.org

715157, 718187, 719197, 734347, 736367, 739397, 748487, 752527, 757577, 760607, 767677, 775757, 779797, 781817, 785857, 796967, 797977, 7100510057, 7102010207, 7103010307, 7104710477, 7105410547, 7106210627, 7106510657, 7107410747

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 13 2005

			715157 is prime and of the form 7nn7 for n=15.
7100510057 is prime and of the form 7nn7 for n=1005.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A102897 for sequence of all numbers of form 7nn7. A102500 for the n values corresponding to the primes in this sequence.

Select[Table[FromDigits[Join[{7},IntegerDigits[n],IntegerDigits[n],{7}]],{n,1100}],PrimeQ] (* Harvey P. Dale, Jun 05 2024 *)

cp's OEIS Frontend

A108800 Number of nonisomorphic systems enumerated by A102895.

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A326875 BII-numbers of set-systems that are closed under union.

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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.

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A326880 BII-numbers of set-systems that are closed under nonempty intersection.

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A326881 Number of set-systems with {} that are closed under intersection and cover n vertices.

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A326906 Number of sets of subsets of {1..n} that are closed under union and cover all n vertices.

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A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

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A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

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