A326702 -id:A326702 - cp's OEIS frontend

A326754 BII-numbers of set-systems covering an initial interval of positive integers.

0, 1, 3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Gus Wiseman, Jul 23 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 sequence of all covering set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  18: {{2},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  26: {{2},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}

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

Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[0,100],normQ[Join@@bpe/@bpe[#]]&]

from itertools import chain, count, islice
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):
        s = set(i for i in chain.from_iterable([bin_i(k) for k in bin_i(n)]))
        y = len(s)
        if sum(s) == (y*(y+1))//2:
            yield n
A326754_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 20 2024

A367908 Numbers n such that there is only one way to choose a different binary index of each binary index of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 56, 67, 69, 70, 73, 74, 81, 88, 98, 104, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152, 154, 156, 162, 163, 165, 166, 168

Offset: 1

Gus Wiseman, Dec 11 2023

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in exactly one way.
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. 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 digits (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 axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The set-system {{1},{1,2},{1,3}} with BII-number 21 satisfies the axiom in exactly one way, namely (1,2,3), so 21 is in the sequence.
The terms together with the corresponding set-systems begin:
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..2000
Wikipedia, Axiom of choice.

These set-systems are counted by A367904.
Positions of 1's in A367905, firsts A367910, sorted firsts A367911.
If there is at least one choice we get A367906, counted by A367902.
If there are no choices we get A367907, counted by A367903.
If there are multiple choices we get A367909, counted by A367772.
The version for MM-numbers of multiset partitions is A368101.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A059201 counts covering T_0 set-systems.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A368098 counts unlabeled multiset partitions for axiom, complement A368097.
Cf. A000612, A059519, A309326, A326675, A326702, A326753, A326872, A355529, A367902, A367912.
BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers), A326783 (uniform), A326784 (regular), A326788 (simple), A330217 (achiral).

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

from itertools import count, islice, product
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen(): #generator of terms
    for n in count(1):
        p = list(product(*[bin_i(k) for k in bin_i(n)]))
        x,c = len(p),0
        for j in range(x):
            if len(set(p[j])) == len(p[j]): c += 1
            if j+1 == x and c == 1: yield(n)
A367908_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Feb 10 2024

A367907 U A367908 U A367909 = A000027.

A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

1, 2, 4, 7, 14, 24, 39, 61, 122, 203, 315, 469, 676, 952, 1307, 1771, 3542, 5708, 8432, 11877, 16123, 21415, 27835, 35757, 45343, 57010, 70778, 87384, 106479, 129304, 155802, 187223, 374446, 588130, 835800, 1124981, 1456282, 1841361, 2281772, 2791896, 3367162

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

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.

			The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {1,3,4}
                         {2,3,4}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A134964, complement A140637.
Simple graphs not of this type are counted by A367867, covering A367868.
Set systems of this type are counted by A367902, ranks A367906.
Set systems not of this type are counted by A367903, ranks A367907.
Set systems uniquely of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
Factorizations are counted by A368414/A370814, complement A368413/A370813.
For prime indices we have A370582, differences A370586.
The complement for prime indices is A370583, differences A370587.
The complement is A370637, differences A370589, without ones A370643.
The case of a unique choice is A370638, maxima A370640, differences A370641.
First differences are A370639.
The minimal case of the complement is A370642, without ones A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]

a(2^n - 1) = A367902(n).

Partial sums of A370639.

a(19)-a(40) from Alois P. Heinz, Mar 09 2024

A326750 BII-numbers of clutters (connected antichains of nonempty sets).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 52, 64, 128, 256, 260, 272, 276, 292, 304, 308, 320, 512, 516, 532, 544, 548, 560, 564, 576, 768, 772, 784, 788, 800, 804, 816, 820, 832, 1024, 1040, 1056, 1072, 1088, 2048, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304

Offset: 1

Gus Wiseman, Jul 23 2019

A binary index of n is any position of a 1 in its reversed binary expansion. 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. In an antichain, no edge is a subset or superset of any other edge.

			The sequence of all clutters together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    8: {{3}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  256: {{1,4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  292: {{1,2},{2,3},{1,4}}
  304: {{1,3},{2,3},{1,4}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  320: {{1,2,3},{1,4}}

John Tyler Rascoe, Table of n, a(n) for n = 1..6834
John Tyler Rascoe, Python program.

The number of clutters spanning n vertices is A048143(n).
Cf. A000120, A001187, A048793, A070939, A072639, A304986, A326031, A326702, A326753.
Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&]

Python
```
# see linked program
```

Intersection of A326749 and A326704.

A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 4, 4, 4, 4, 7, 7, 7, 7, 3, 3, 3, 3, 5, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 8, 8, 8, 8

Offset: 0

Gus Wiseman, Dec 12 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

The run-lengths are all 4 or 8.

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with multiset choices {1,1,2}, {1,1,3}, {1,2,2}, {1,2,3}, {1,3,3}, {2,2,3}, {2,3,3}, so a(52) = 7.

Positions of ones are A253317.
The version for multisets and divisors is A355733, for sequences A355731.
The version for multisets is A355744, for sequences A355741.
For a sequence of distinct choices we have A367905, firsts A367910.
Positions of first appearances are A367913, sorted A367915.
Choosing a sequence instead of multiset gives A368109, firsts A368111.
Choosing a set instead of multiset gives A368183, firsts A368184.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355735, A355739, A355740, A355745, A367771, A367906.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]], {n,0,100}]

A326703 BII-numbers of chains of nonempty sets.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 24, 32, 34, 40, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 256, 257, 384, 512, 514, 640, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1408, 1536, 1538, 1664, 2048, 2056, 2176, 4096, 4097, 4104, 4112, 4113, 4120

Offset: 1

Gus Wiseman, Jul 21 2019

A binary index of n is any position of a 1 in its reversed binary expansion. 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, it follows that the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. In a chain of sets, every edge is a subset or superset of every other edge.

			The sequence of all chains of nonempty sets together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    5: {{1},{1,2}}
    6: {{2},{1,2}}
    8: {{3}}
   16: {{1,3}}
   17: {{1},{1,3}}
   24: {{3},{1,3}}
   32: {{2,3}}
   34: {{2},{2,3}}
   40: {{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}}
   72: {{3},{1,2,3}}
   80: {{1,3},{1,2,3}}
   81: {{1},{1,3},{1,2,3}}
   88: {{3},{1,3},{1,2,3}}
   96: {{2,3},{1,2,3}}
   98: {{2},{2,3},{1,2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..4860

Chains of nonempty sets are counted by A000629.
MM-numbers of chains of multisets are A318991.
BII-numbers of antichains of nonempty sets are A326704.
Cf. A000120, A014466, A029931, A035327, A048793, A070939, A326031, A326701, A326702.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,100],stableQ[bpe/@bpe[#],!SubsetQ[#1,#2]&&!SubsetQ[#2,#1]&]&]

from itertools import chain, count, combinations, islice
from sympy.combinatorics.subsets import ksubsets
def subsets(x):
    for i in range(1,len(x)):
        for j in ksubsets(x,i):
            yield(list(j))
def a_gen(): #generator of terms
    yield 0
    for n in count(1):
        t,v,j = [[]],[],0
        for i in chain.from_iterable(combinations(range(1, n+1), r) for r in range(n+1)):
            if n in i:
                t[j].append([list(i)])
        while n:
            t.append([])
            for i in t[j]:
                if len(i[-1]) > 1:
                    for k in list(subsets(i[-1])):
                        t[j+1].append(i.copy()+[k])
            if len(t[j+1]) < 1:
                break
            j += 1
        for j in chain.from_iterable(t):
            v.append(sum(2**(sum(2**(m-1) for m in k)-1) for k in j))
        yield from sorted(v)
A326703_list = list(islice(a_gen(), 55)) # John Tyler Rascoe, Jun 07 2024

A327041 a(n) is the number whose binary indices are the union of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 3, 3, 4, 5, 6, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Gus Wiseman, Aug 19 2019

nonn

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 set-system 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.

			22 is the BII-number of {{2},{1,2},{1,3}}, and 7 has binary indices {1,2,3}, so a(22) = 7.

Tilman Piesk, Illustration of the first 128 terms

Indices of records are A253317.

Cf. A000120, A001511, A029931, A035327, A048793, A070939, A072639, A326031, A326702, A326947, A261283 (XOR equivalent).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[2^Union@@bpe/@bpe[n]]/2,{n,0,100}]

A330099 BII-numbers of brute-force normalized set-systems.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 15, 19, 20, 21, 23, 31, 33, 37, 51, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 79, 83, 84, 85, 87, 95, 97, 101, 115, 116, 117, 119, 127, 139, 143, 159, 191, 203, 207, 223, 255, 267, 271, 275, 276, 277, 279, 287, 307, 308, 309, 311, 319, 331

Offset: 1

Gus Wiseman, Dec 02 2019

First differs from A330100 in having 545 and lacking 179, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 179: {{1},{2},{4},{1,3},{2,3}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the brute-force normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of sets is first by length and then lexicographically.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}
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 set-system 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.
There are A055621(n) entries m such that A326702(m) = n, where A326702(k) is the number of covered vertices in the set-system with BII-number k.
There are A283877(n) entries m such that A326031(m) = n, where A326031(k) is the weight of the set-system with BII-number k.

			The sequence of all nonempty brute-force normalized set-systems together with their BII-numbers begins:
   1: {1}                  52: {12}{13}{23}
   3: {1}{2}               53: {1}{12}{13}{23}
   4: {12}                 55: {1}{2}{12}{13}{23}
   5: {1}{12}              63: {1}{2}{3}{12}{13}{23}
   7: {1}{2}{12}           64: {123}
  11: {1}{2}{3}            65: {1}{123}
  15: {1}{2}{3}{12}        67: {1}{2}{123}
  19: {1}{2}{13}           68: {12}{123}
  20: {12}{13}             69: {1}{12}{123}
  21: {1}{12}{13}          71: {1}{2}{12}{123}
  23: {1}{2}{12}{13}       75: {1}{2}{3}{123}
  31: {1}{2}{3}{12}{13}    79: {1}{2}{3}{12}{123}
  33: {1}{23}              83: {1}{2}{13}{123}
  37: {1}{12}{23}          84: {12}{13}{123}
  51: {1}{2}{13}{23}       85: {1}{12}{13}{123}

Equals the image/fixed points of the idempotent sequence A330101.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems by span are A055621.
Unlabeled spanning set-systems by weight are A283877.
Cf. A000612, A300913, A321405, A330061, A330102, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[0,100],Sort[bpe/@bpe[#]]==brute[bpe/@bpe[#]]&]

A309326 BII-numbers of minimal covers.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 320, 512, 513, 516, 520, 521, 524, 528, 544, 545, 548

Offset: 1

Gus Wiseman, Jul 23 2019

nonn

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. A minimal cover is a set-system where every edge contains at least one vertex that does not belong to any other edge.

			The sequence of all minimal covers together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    4: {{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   12: {{1,2},{3}}
   16: {{1,3}}
   18: {{2},{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   33: {{1},{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}

Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],And@@Table[Union@@Delete[bpe/@bpe[#],i]!=Union@@bpe/@bpe[#],{i,Length[bpe/@bpe[#]]}]&]

A253317 Indices in A261283 where records occur.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 10, 11, 128, 129, 130, 131, 136, 137, 138, 139, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906, 32907, 2147483648, 2147483649, 2147483650, 2147483651, 2147483656, 2147483657

Offset: 1

Philippe Beaudoin, Dec 30 2014

From Gus Wiseman, Dec 29 2023: (Start)
These are numbers whose binary indices are all powers of 2, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, the terms together with their binary expansions and binary indices begin:
       0 ~ {}
       1 ~ {1}
      10 ~ {2}
      11 ~ {1,2}
    1000 ~ {4}
    1001 ~ {1,4}
    1010 ~ {2,4}
    1011 ~ {1,2,4}
10000000 ~ {8}
10000001 ~ {1,8}
10000010 ~ {2,8}
10000011 ~ {1,2,8}
10001000 ~ {4,8}
10001001 ~ {1,4,8}
10001010 ~ {2,4,8}
10001011 ~ {1,2,4,8}
For powers of 3 we have A368531.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..4096
Lorenzo Sauras-Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.

Cf. A053644 (most significant bit).
A048793 lists binary indices, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A058891, A062050, A072639, A326031, A326675, A326702, A367771, A367912, A368183, A368109, A368531.

a := proc(n) local k, A:
A := [seq(0,i=1..n)]: A[1]:=0:
for k from 1 to n-1 do
   A[k+1] := A[k-2^ilog2(k)+1]+2^(2^ilog2(k)-1): od:
return A[n]: end proc: # Lorenzo Sauras Altuzarra, Dec 18 2019
# second Maple program:
a:= n-> (l-> add(l[i+1]*2^(2^i-1), i=0..nops(l)-1))(Bits[Split](n-1)):
seq(a(n), n=1..38);  # Alois P. Heinz, Dec 13 2023

Nest[Append[#1, #1[[-#2]] + 2^(#2 - 1)] & @@ {#, 2^(IntegerLength[Length[#], 2] - 1)} &, {0, 1}, 36] (* Michael De Vlieger, May 08 2020 *)

a(n)={if(n<=1, 0, my(t=1<Andrew Howroyd, Dec 20 2019

a(1) = 0 and a(n) = a(n-A053644(n-1)) + 2^(A053644(n-1)-1). - Lorenzo Sauras Altuzarra, Dec 18 2019
a(n) = A358126(n-1) / 2. - Tilman Piesk, Dec 18 2022
a(2^n+1) = 2^(2^n-1) = A058891(n+1). - Gus Wiseman, Dec 29 2023
a(2^n) = A072639(n). - Gus Wiseman, Dec 29 2023
G.f.: 1/(1-x) * Sum_{k>=0} (2^(-1+2^k))*x^2^k/(1+x^2^k). - John Tyler Rascoe, May 22 2024

Corrected reference in name from A253315 to A261283. - Tilman Piesk, Dec 18 2022

cp's OEIS Frontend

A326754 BII-numbers of set-systems covering an initial interval of positive integers.

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A367908 Numbers n such that there is only one way to choose a different binary index of each binary index of n.

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A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

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A326750 BII-numbers of clutters (connected antichains of nonempty sets).

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A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

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A326703 BII-numbers of chains of nonempty sets.

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A327041 a(n) is the number whose binary indices are the union of the set-system with BII-number n.

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A330099 BII-numbers of brute-force normalized set-systems.

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