A326753 -id:A326753 - cp's OEIS frontend

A329627 Smallest BII-number of a clutter (connected antichain) with n edges.

0, 1, 20, 52, 308, 820, 2868, 68404, 199476, 723764

Offset: 0

Gus Wiseman, Nov 28 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 set-system (finite set of finite nonempty sets of positive integers) 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 set-system is an antichain if no edge is a proper subset of any other.
For n > 1, a(n) appears to be the number whose binary indices are the first n terms of A018900.

			The sequence of terms together with their corresponding set-systems begins:
       0: {}
       1: {{1}}
      20: {{1,2},{1,3}}
      52: {{1,2},{1,3},{2,3}}
     308: {{1,2},{1,3},{2,3},{1,4}}
     820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
    2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
   68404: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5}}
  199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
  723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}

The version for MM-numbers is A329555.
BII-numbers of clutters are A326750.
Clutters of sets are counted by A048143.
Minimum BII-numbers of connected set-systems are A329625.
Minimum BII-numbers of antichains are A329626.
MM-numbers of connected weak antichains of multisets are A329559.
Cf. A048793, A070939, A072639, A320275, A322113, A326031, A326704, A326753, A329628, A329632.

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}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&&Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]

A371450 MM-number of the set-system with BII-number n.

Original entry on oeis.org

1, 3, 5, 15, 13, 39, 65, 195, 11, 33, 55, 165, 143, 429, 715, 2145, 29, 87, 145, 435, 377, 1131, 1885, 5655, 319, 957, 1595, 4785, 4147, 12441, 20735, 62205, 47, 141, 235, 705, 611, 1833, 3055, 9165, 517, 1551, 2585, 7755, 6721, 20163, 33605, 100815, 1363, 4089

Offset: 0

Gus Wiseman, Apr 02 2024

nonn

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}}.

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 (finite set of finite nonempty sets of positive integers) 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 set-system with BII-number 30 is {{2},{1,2},{3},{1,3}} with MM-number prime(3) * prime(6) * prime(5) * prime(10) = 20735.
The terms together with their prime indices and binary indices of prime indices begin:
     1 -> {}        -> {}
     3 -> {2}       -> {{1}}
     5 -> {3}       -> {{2}}
    15 -> {2,3}     -> {{1},{2}}
    13 -> {6}       -> {{1,2}}
    39 -> {2,6}     -> {{1},{1,2}}
    65 -> {3,6}     -> {{2},{1,2}}
   195 -> {2,3,6}   -> {{1},{2},{1,2}}
    11 -> {5}       -> {{3}}
    33 -> {2,5}     -> {{1},{3}}
    55 -> {3,5}     -> {{2},{3}}
   165 -> {2,3,5}   -> {{1},{2},{3}}
   143 -> {5,6}     -> {{1,2},{3}}
   429 -> {2,5,6}   -> {{1},{1,2},{3}}
   715 -> {3,5,6}   -> {{2},{1,2},{3}}
  2145 -> {2,3,5,6} -> {{1},{2},{1,2},{3}}

The sorted version is A329629, with empties A302494.
A019565 gives Heinz number of binary indices.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A326753 counts connected components for BII-numbers, ones A326749.
Cf. A000720, A003963, A087086, A096111, A275024, A302242, A302505, A302521, A326782, A329557, A329630, A368109.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Times@@Prime/@(Times@@Prime/@#&/@bix/@bix[n]),{n,0,30}]

A327110 BII-numbers of set-systems with spanning edge-connectivity 3.

Original entry on oeis.org

116, 117, 118, 119, 124, 125, 126, 127, 1796, 1797, 1798, 1799, 1904, 1905, 1906, 1907, 1908, 1909, 1910, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1918, 1919, 1924, 1925, 1926, 1927, 2032, 2033, 2034, 2035, 2036, 2037, 2038, 2039, 2040, 2041, 2042, 2043, 2044

Offset: 1

Gus Wiseman, Oct 03 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 (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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			The sequence of all set-systems with spanning edge-connectivity 3 together with their BII-numbers begins:
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   117: {{1},{1,2},{1,3},{2,3},{1,2,3}}
   118: {{2},{1,2},{1,3},{2,3},{1,2,3}}
   119: {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
   124: {{1,2},{3},{1,3},{2,3},{1,2,3}}
   125: {{1},{1,2},{3},{1,3},{2,3},{1,2,3}}
   126: {{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
   127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
  1796: {{1,2},{1,4},{2,4},{1,2,4}}
  1797: {{1},{1,2},{1,4},{2,4},{1,2,4}}
  1798: {{2},{1,2},{1,4},{2,4},{1,2,4}}
  1799: {{1},{2},{1,2},{1,4},{2,4},{1,2,4}}
  1904: {{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1905: {{1},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1906: {{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1907: {{1},{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1908: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1909: {{1},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1910: {{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
  1911: {{1},{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}

Positions of 3's in A327144.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A001187, A095983, A322395, A326749, A326753, A326787, A327069, A327071, A327103, A327130, A327146, A327147.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[1000],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==3&]

A327376 BII-numbers of set-systems with vertex-connectivity 3.

Original entry on oeis.org

2868, 2869, 2870, 2871, 2876, 2877, 2878, 2879, 2880, 2881, 2882, 2883, 2884, 2885, 2886, 2887, 2888, 2889, 2890, 2891, 2892, 2893, 2894, 2895, 2896, 2897, 2898, 2899, 2900, 2901, 2902, 2903, 2904, 2905, 2906, 2907, 2908, 2909, 2910, 2911, 2912, 2913, 2914

Offset: 1

Gus Wiseman, Sep 05 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 (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 vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			The sequence of all set-systems with vertex-connectivity 3 together with their BII-numbers begins:
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2869: {{1},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2870: {{2},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2871: {{1},{2},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2876: {{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2877: {{1},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2878: {{2},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2879: {{1},{2},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2880: {{1,2,3},{1,4},{2,4},{3,4}}
  2881: {{1},{1,2,3},{1,4},{2,4},{3,4}}
  2882: {{2},{1,2,3},{1,4},{2,4},{3,4}}
  2883: {{1},{2},{1,2,3},{1,4},{2,4},{3,4}}
  2884: {{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2885: {{1},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2886: {{2},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2887: {{1},{2},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2888: {{3},{1,2,3},{1,4},{2,4},{3,4}}
  2889: {{1},{3},{1,2,3},{1,4},{2,4},{3,4}}
  2890: {{2},{3},{1,2,3},{1,4},{2,4},{3,4}}
  2891: {{1},{2},{3},{1,2,3},{1,4},{2,4},{3,4}}

Positions of 3's in A327051.
BII-numbers for vertex-connectivity 2 are A327374.
BII-numbers for spanning edge-connectivity >= 3 are A327110.
The enumeration of labeled graphs by vertex-connectivity is A327334.
Cf. A000120, A013922, A048793, A070939, A259862, A323818, A326031, A326753, A326786, A327375.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
Select[Range[0,3000],vertConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]==3&]

A330296 BII-numbers of set partitions with at least two blocks.

Original entry on oeis.org

3, 9, 10, 11, 12, 18, 33, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 160, 161, 192, 258, 264, 266, 288, 513, 520, 521, 528, 1032, 2049, 2050, 2051, 2052, 4098, 8193, 32769, 32770, 32771, 32772, 32776, 32777, 32778, 32779, 32780, 32784, 32786, 32800

Offset: 1

Gus Wiseman, Dec 10 2019

nonn

			The sequence of all set partitions with at least two parts together with their BII-numbers begins:
    3: {1}{2}          140: {3}{4}{12}     2049: {1}{34}
    9: {1}{3}          144: {4}{13}        2050: {2}{34}
   10: {2}{3}          146: {2}{4}{13}     2051: {1}{2}{34}
   11: {1}{2}{3}       160: {4}{23}        2052: {12}{34}
   12: {3}{12}         161: {1}{4}{23}     4098: {2}{134}
   18: {2}{13}         192: {4}{123}       8193: {1}{234}
   33: {1}{23}         258: {2}{14}       32769: {1}{5}
  129: {1}{4}          264: {3}{14}       32770: {2}{5}
  130: {2}{4}          266: {2}{3}{14}    32771: {1}{2}{5}
  131: {1}{2}{4}       288: {14}{23}      32772: {5}{12}
  132: {4}{12}         513: {1}{24}       32776: {3}{5}
  136: {3}{4}          520: {3}{24}       32777: {1}{3}{5}
  137: {1}{3}{4}       521: {1}{3}{24}    32778: {2}{3}{5}
  138: {2}{3}{4}       528: {13}{24}      32779: {1}{2}{3}{5}
  139: {1}{2}{3}{4}   1032: {3}{124}      32780: {3}{5}{12}

BII-numbers of set partitions are A326701.

Cf. A048793, A070939, A072639, A326031, A326753, A329661.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],Length[bpe[#]]>=2&&Length[Join@@bpe/@bpe[#]]==Length[Union@@bpe/@bpe[#]]&]

Equal the complement of A000079 in A326701.

cp's OEIS Frontend

A329627 Smallest BII-number of a clutter (connected antichain) with n edges.

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A371450 MM-number of the set-system with BII-number n.

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A327110 BII-numbers of set-systems with spanning edge-connectivity 3.

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A327376 BII-numbers of set-systems with vertex-connectivity 3.

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A330296 BII-numbers of set partitions with at least two blocks.

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