A329625 Smallest BII-number of a connected set-system with n edges.

0, 1, 5, 7, 23, 31, 63, 127, 383, 511, 1023, 2047, 4095, 8191

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..13.

Example

The sequence of terms together with their corresponding set-systems begins:
     0: {}
     1: {{1}}
     5: {{1},{1,2}}
     7: {{1},{2},{1,2}}
    23: {{1},{2},{1,2},{1,3}}
    31: {{1},{2},{1,2},{3},{1,3}}
    63: {{1},{2},{1,2},{3},{1,3},{2,3}}
   127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
   383: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{1,4}}
   511: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4}}

Mathematica

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]]]]]]]]];
First/@GatherBy[Select[Range[0, 1000], Length[csm[bpe/@bpe[#]]]<=1&], Length[bpe[#]]&]

Crossrefs

The smallest BII-number of a set-system with n edges is A000225(n).

The smallest BII-number of a set-system with n vertices is A072639(n).

BII-numbers of connected set-systems are A326749.

MM-numbers of connected set-systems are A328514.

The case of clutters is A329627.

Cf. A048143, A048793, A070939, A304716, A305078, A326031, A326753, A329552, A329553, A329626.

Sequence in context: A028281 A240102 A105890 * A018656 A214520 A156123

Adjacent sequences: A329622 A329623 A329624 * A329626 A329627 A329628

Keyword

nonn,more

Author

Gus Wiseman, Nov 28 2019

Status

approved