A326701 -id:A326701 - cp's OEIS frontend

A327106 BII-numbers of set-systems with maximum degree 2.

5, 6, 7, 13, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27, 28, 30, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 65, 66, 67, 68, 72, 73, 74, 75, 76, 80, 82, 96, 97, 133, 134, 135, 141, 142, 143, 145, 147, 148, 150, 152, 153, 154, 155, 156, 158, 162

Offset: 1

Gus Wiseman, Aug 26 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.

In a set-system, the degree of a vertex is the number of edges containing it.

			The sequence of all set-systems with maximum degree 2 together with their BII-numbers begins:
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  26: {{2},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}

Positions of 2's in A327104.
Graphs with maximum degree 2 are counted by A136284.
Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326786, A327041, A327103.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],If[#==0,0,Max@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]]==2&]

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

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 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

A327106 BII-numbers of set-systems with maximum degree 2.

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