A330217 -id:A330217 - cp's OEIS frontend

A367905 Number of ways to choose a sequence of different binary indices, one of each binary index of n.

1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 1, 3, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 2, 2, 1, 4, 1, 1, 0, 2, 1, 1, 0, 2, 0, 0, 0, 4, 1, 2, 0, 3, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 10 2023

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}.

			352 has binary indices of binary indices {{2,3},{1,2,3},{1,4}}, and there are six possible choices (2,1,4), (2,3,1), (2,3,4), (3,1,4), (3,2,1), (3,2,4), so a(352) = 6.

John Tyler Rascoe, Table of n, a(n) for n = 0..16384
Wikipedia, Axiom of choice.

A version for multisets is A367771, see A355529, A355740, A355744, A355745.
Positions of positive terms are A367906.
Positions of zeros are A367907.
Positions of ones are A367908.
Positions of terms > 1 are A367909.
Positions of first appearances are A367910, sorted A367911.
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. A000612, A055621, A072639, A309326, A326031, A326675, A326702, A326753, A367902, A367903, A367904, 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];
Table[Length[Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]],{n,0,100}]

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(0):
        c = 0
        for j in list(product(*[bin_i(k) for k in bin_i(n)])):
            if len(set(j)) == len(j):
                c += 1
        yield c
A367905_list = list(islice(a_gen(), 90)) # John Tyler Rascoe, May 22 2024

A367907 Numbers n such that it is not possible to choose a different binary index of each binary index of n.

Original entry on oeis.org

7, 15, 23, 25, 27, 29, 30, 31, 39, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121

Offset: 1

Gus Wiseman, Dec 11 2023

Also BII-numbers of set-systems (sets of nonempty sets) contradicting a strict version of the axiom of choice.
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},{2},{1,2},{1,3}} with BII-number 23 has choices (1,2,1,1), (1,2,1,3), (1,2,2,1), (1,2,2,3), but none of these has all different elements, so 23 is in the sequence.
The terms together with the corresponding set-systems begin:
   7: {{1},{2},{1,2}}
  15: {{1},{2},{1,2},{3}}
  23: {{1},{2},{1,2},{1,3}}
  25: {{1},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  39: {{1},{2},{1,2},{2,3}}
  42: {{2},{3},{2,3}}
  43: {{1},{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  46: {{2},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}

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

These set-systems are counted by A367903, non-isomorphic A368094.
Positions of zeros in A367905, firsts A367910, sorted A367911.
The complement is A367906.
If there is one unique choice we get A367908, counted by A367904.
If there are multiple choices we get A367909, counted by A367772.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
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, A055621, A072639, A083323, A309326, A326702, A326753, A367769, A367901, 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], Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]=={}&]

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 = len(p)
        for j in range(x):
            if len(set(p[j])) == len(p[j]): break
            if j+1 == x: yield(n)
A367907_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Feb 10 2024

A367907 U A367908 U A367909 = A000027.

A367906 Numbers k such that it is possible to choose a different binary index of each binary index of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 48, 49, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 80, 81, 82, 84, 88, 96, 97, 98, 100, 104, 112, 128, 129, 130, 131, 132

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.
A binary index of k (row k 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 k to be obtained by taking the binary indices of each binary index of k. 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 {{2,3},{1,2,3},{1,4}} with BII-number 352 has choices such as (2,1,4) that satisfy the axiom, so 352 is in the sequence.
The terms together with the corresponding set-systems begin:
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  16: {{1,3}}
  17: {{1},{1,3}}

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

These set-systems are counted by A367902, non-isomorphic A368095.
Positions of positive terms in A367905, firsts A367910, sorted A367911.
The complement is A367907.
If there is one unique choice we get A367908, counted by A367904.
If there are multiple choices we get A367909, counted by A367772.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
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, A055621, A059519, A072639, A083323, A309326, A326702, A326753, A367770, 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], Select[Tuples[bpe/@bpe[#]], UnsameQ@@#&]!={}&]

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):
        for j in list(product(*[bin_i(k) for k in bin_i(n)])):
            if len(set(j)) == len(j):
                yield(n); break
A367906_list = list(islice(a_gen(),100)) # John Tyler Rascoe, Dec 23 2023

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.

A083323 a(n) = 3^n - 2^n + 1.

Original entry on oeis.org

1, 2, 6, 20, 66, 212, 666, 2060, 6306, 19172, 58026, 175100, 527346, 1586132, 4766586, 14316140, 42981186, 129009092, 387158346, 1161737180, 3485735826, 10458256052, 31376865306, 94134790220, 282412759266, 847255055012

Offset: 0

Paul Barry, Apr 27 2003

Binomial transform of A000225 (if this starts 1,1,3,7....).
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 1) x = y. - Ross La Haye, Jan 10 2008
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009

			From _Gus Wiseman_, Dec 10 2019: (Start)
Also the number of achiral set-systems on n vertices, where a set-system is achiral if it is not changed by any permutation of the covered vertices. For example, the a(0) = 1 through a(3) = 20 achiral set-systems are:
  0  0    0           0
     {1}  {1}         {1}
          {2}         {2}
          {12}        {3}
          {1}{2}      {12}
          {1}{2}{12}  {13}
                      {23}
                      {123}
                      {1}{2}
                      {1}{3}
                      {2}{3}
                      {1}{2}{3}
                      {1}{2}{12}
                      {1}{3}{13}
                      {2}{3}{23}
                      {12}{13}{23}
                      {1}{2}{3}{123}
                      {12}{13}{23}{123}
                      {1}{2}{3}{12}{13}{23}
                      {1}{2}{3}{12}{13}{23}{123}
BII-numbers of these set-systems are A330217. Fully chiral set-systems are A330282, with covering case A330229.
(End)

M. H. Albert, M. D. Atkinson, and V. Vatter, Inflations of geometric grid classes: three case studies, arXiv:1209.0425 [math.CO], 2012.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A134319, A028243, A000079.

Cf. A000612, A003238, A330098, A330234.

List([0..30], n -> 3^n-2^n+1); # G. C. Greubel, Feb 13 2019

[3^n-2^n+1: n in [0..30]]; // G. C. Greubel, Feb 13 2019

LinearRecurrence[{6,-11,6}, {1,2,6}, 30] (* G. C. Greubel, Feb 13 2019 *)

a(n)=3^n-2^n+1 \\ Charles R Greathouse IV, Oct 07 2015

[3^n-2^n+1 for n in range(30)] # G. C. Greubel, Feb 13 2019

G.f.: (1-4*x+5*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f.: exp(3*x) - exp(2*x) + exp(x).
Row sums of triangle A134319. - Gary W. Adamson, Oct 19 2007
a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 10 2008
a(n) = Sum_{k=0..n}(binomial(n,k)*A255047(k)). - Yuchun Ji, Feb 23 2019

A330223 Number of non-isomorphic achiral multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 5, 12, 9, 30, 17, 52, 44, 94, 58, 211, 103, 302, 242, 552, 299, 1024, 492, 1592, 1007, 2523, 1257, 4636, 2000, 6661, 3705, 10823, 4567, 18147, 6844, 26606, 12272, 40766, 15056, 67060, 21639, 95884, 37357, 146781, 44585, 230098, 63263, 330889, 106619, 491182, 124756

Offset: 0

Gus Wiseman, Dec 07 2019

A multiset partition is a finite multiset of finite nonempty multisets. It is achiral if it is not changed by any permutation of the vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}
       {12}    {123}      {1122}        {12345}
       {1}{1}  {1}{11}    {1234}        {1}{1111}
       {1}{2}  {1}{1}{1}  {1}{111}      {11}{111}
               {1}{2}{3}  {11}{11}      {1}{1}{111}
                          {11}{22}      {1}{11}{11}
                          {12}{12}      {1}{1}{1}{11}
                          {1}{1}{11}    {1}{1}{1}{1}{1}
                          {1}{2}{12}    {1}{2}{3}{4}{5}
                          {1}{1}{1}{1}
                          {1}{1}{2}{2}
                          {1}{2}{3}{4}
Non-isomorphic representatives of the a(6) = 30 multiset partitions:
  {111111}  {1}{11111}  {1}{1}{1111}  {1}{1}{1}{111}  {1}{1}{1}{1}{11}
  {111222}  {11}{1111}  {1}{11}{111}  {1}{1}{11}{11}  {1}{1}{2}{2}{12}
  {112233}  {111}{111}  {11}{11}{11}  {1}{2}{11}{22}
  {123456}  {111}{222}  {11}{12}{22}  {1}{2}{12}{12}
            {112}{122}  {11}{22}{33}  {1}{2}{3}{123}    {1}{1}{1}{1}{1}{1}
            {12}{1122}  {1}{2}{1122}                    {1}{1}{1}{2}{2}{2}
            {123}{123}  {12}{12}{12}                    {1}{1}{2}{2}{3}{3}
                        {12}{13}{23}                    {1}{2}{3}{4}{5}{6}

Planted achiral trees are A003238.
Achiral set-systems are counted by A083323.
BII-numbers of achiral set-systems are A330217.
Achiral integer partitions are counted by A330224.
Non-isomorphic fully chiral multiset partitions are A330227.
MM-numbers of achiral multisets of multisets are A330232.
Achiral factorizations are A330234.
Cf. A001055, A007716, A283877, A317533, A330098, A330233.

a(10)-a(11) and a(13) from Erich Friedman, Nov 20 2024
a(12) from Bert Dobbelaere, Apr 29 2025
More terms from Bert Dobbelaere, May 02 2025

A330232 MM-numbers of achiral multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80

Offset: 1

Gus Wiseman, Dec 08 2019

nonn

First differs from A322554 in lacking 141.
A multiset of multisets is achiral if it is not changed by any permutation of the vertices.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of non-achiral multisets of multisets (the complement of this sequence) together with their MM-numbers begins:
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  45: {{1},{1},{2}}
  61: {{1,2,2}}
  65: {{2},{1,2}}
  69: {{1},{2,2}}
  70: {{},{2},{1,1}}
  71: {{1,1,3}}
  74: {{},{1,1,2}}
  75: {{1},{2},{2}}
  77: {{1,1},{3}}
  78: {{},{1},{1,2}}
  87: {{1},{1,3}}
  89: {{1,1,1,2}}
  90: {{},{1},{1},{2}}

The fully-chiral version is A330236.
Achiral set-systems are counted by A083323.
MG-numbers of planted achiral trees are A214577.
MM-weight is A302242.
MM-numbers of costrict (or T_0) multisets of multisets are A322847.
BII-numbers of achiral set-systems are A330217.
Non-isomorphic achiral multiset partitions are A330223.
Achiral integer partitions are counted by A330224.
Achiral factorizations are A330234.
Cf. A001055, A003238, A007716, A056239, A112798, A303975, A330098, A330230, A330233.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]]
Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==1&]

A330231 Number of distinct set-systems that can be obtained by permuting the vertices of the set-system with BII-number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 6, 6, 3, 1, 2, 3, 6, 3, 3, 6, 6, 2, 1, 6, 3, 6, 6, 3, 3, 1, 3, 2, 6, 3, 6, 3, 6, 2, 6, 1, 3, 6, 3, 6, 3, 3, 6, 6, 3, 1, 3, 3, 3, 3, 6, 6, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 6, 6, 3, 3, 3, 3, 1, 3, 6, 6, 3, 3, 6, 3, 6, 3, 3, 6

Offset: 0

Gus Wiseman, Dec 09 2019

nonn

A set-system is a finite set of finite nonempty sets.

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.

			30 is the MM-number of {{2},{3},{1,2},{1,3}}, with vertex permutations
  {{1},{2},{1,3},{2,3}}
  {{1},{3},{1,2},{2,3}}
  {{2},{3},{1,2},{1,3}}
so a(30) = 3.

Positions of 1's are A330217.
Positions of first appearances are A330218.
The version for MM-numbers is A330098.
Achiral set-systems are counted by A083323.
BII-numbers of fully chiral set-systems are A330226.
Cf. A000120, A003238, A007716, A016031, A048793, A055621, A070939, A214577, A326031, A326702, A330101, A330195, A330229, A330230, A330233.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[graprms[bpe/@bpe[n]]],{n,0,100}]

a(n) is a divisor of A326702(n)!.

A367909 Numbers n such that there is more than one way to choose a different binary index of each binary index of n.

Original entry on oeis.org

4, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 65, 66, 68, 72, 76, 80, 82, 84, 96, 97, 100, 112, 132, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 193, 194, 196, 200, 204, 208, 210, 212, 224, 225, 228, 240, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288

Offset: 1

Gus Wiseman, Dec 11 2023

nonn

Also BII-numbers of set-systems (sets of nonempty sets) satisfying a strict version of the axiom of choice in more than 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 only one way (1,2,3), so 21 is not in the sequence.
The terms together with the corresponding set-systems begin:
   4: {{1,2}}
  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}}
  52: {{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Wikipedia, Axiom of choice.

These set-systems are counted by A367772.
Positions of terms > 1 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 is one unique choice we get A367908, counted by A367904.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
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 per axiom, complement A368097.
Cf. A000612, A055621, A072639, A309326, A326702, A326753, A355529, A368100.
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&]

A367907 U A367908 U A367909 = A000027.

A330226 BII-numbers of fully chiral set-systems.

Original entry on oeis.org

0, 1, 2, 5, 6, 8, 13, 14, 17, 19, 22, 23, 24, 26, 28, 29, 34, 35, 37, 39, 40, 41, 44, 46, 49, 50, 57, 58, 69, 70, 77, 78, 81, 83, 86, 87, 88, 90, 92, 93, 98, 99, 101, 103, 104, 105, 108, 110, 113, 114, 121, 122, 128, 133, 134, 145, 150, 151, 152, 156, 157, 162

Offset: 1

Gus Wiseman, Dec 08 2019

nonn

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.

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.

			The sequence of all fully chiral set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{3},{1,2},{1,3}}
  29: {{1},{3},{1,2},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}
  39: {{1},{2},{1,2},{2,3}}
For example, 28 is in the sequence because all six permutations give different representatives, namely:
  {{1},{1,2},{2,3}}
  {{1},{1,3},{2,3}}
  {{2},{1,2},{1,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,2},{1,3}}
  {{3},{1,2},{2,3}}

A subset of A326947.
Achiral set-systems are counted by A083323.
BII-numbers of achiral set-systems are A330217.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral partitions are counted by A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000120, A000612, A016031, A048793, A070939, A326031, A326702, A330218, A330231, A330232.

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

cp's OEIS Frontend

A367905 Number of ways to choose a sequence of different binary indices, one of each binary index of n.

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A367907 Numbers n such that it is not possible to choose a different binary index of each binary index of n.

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A367906 Numbers k such that it is possible to choose a different binary index of each binary index of k.

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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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A083323 a(n) = 3^n - 2^n + 1.

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A330223 Number of non-isomorphic achiral multiset partitions of weight n.

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A330232 MM-numbers of achiral multisets of multisets.

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A330231 Number of distinct set-systems that can be obtained by permuting the vertices of the set-system with BII-number n.