

A327105


BIInumbers of setsystems with minimum degree 1.


11



1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 48, 49, 50, 56, 57, 58, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 88, 89, 96, 98, 104, 106, 128
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem (finite set of finite nonempty sets) has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.
In a setsystem, the degree of a vertex is the number of edges containing it.


LINKS

Table of n, a(n) for n=1..66.


EXAMPLE

The sequence of all setsystems with minimum degree 1 together with their BIInumbers begins:
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}}
15: {{1},{2},{1,2},{3}}
16: {{1,3}}
17: {{1},{1,3}}
18: {{2},{1,3}}
19: {{1},{2},{1,3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}


MATHEMATICA

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


CROSSREFS

Positions of 1's in A327103.
BIInumbers for minimum degree > 1 are A327107.
Graphs with minimum degree 1 are counted by A245797, with covering case A327227.
Setsystems with minimum degree 1 are counted by A327228, with covering case A327229.
Cf. A000120, A029931, A048793, A058891, A070939, A326031, A326701, A326786, A327041, A327104, A327230.
Sequence in context: A194866 A004726 A298747 * A129618 A038673 A183219
Adjacent sequences: A327102 A327103 A327104 * A327106 A327107 A327108


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 26 2019


STATUS

approved



