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A327107
BII-numbers of set-systems with minimum vertex-degree > 1.
5
7, 25, 30, 31, 42, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 99, 100, 101, 102, 103, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124
OFFSET
1,1
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) 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.
EXAMPLE
The sequence of all set-systems with maximum degree > 1 together with their BII-numbers begins:
7: {{1},{2},{1,2}}
25: {{1},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
31: {{1},{2},{1,2},{3},{1,3}}
42: {{2},{3},{2,3}}
45: {{1},{1,2},{3},{2,3}}
47: {{1},{2},{1,2},{3},{2,3}}
51: {{1},{2},{1,3},{2,3}}
52: {{1,2},{1,3},{2,3}}
53: {{1},{1,2},{1,3},{2,3}}
54: {{2},{1,2},{1,3},{2,3}}
55: {{1},{2},{1,2},{1,3},{2,3}}
59: {{1},{2},{3},{1,3},{2,3}}
60: {{1,2},{3},{1,3},{2,3}}
61: {{1},{1,2},{3},{1,3},{2,3}}
62: {{2},{1,2},{3},{1,3},{2,3}}
63: {{1},{2},{1,2},{3},{1,3},{2,3}}
75: {{1},{2},{3},{1,2,3}}
76: {{1,2},{3},{1,2,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[100], Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]>1&]
CROSSREFS
Positions of terms > 1 in A327103.
BII-numbers for minimum degree 1 are A327105.
Graphs with minimum degree > 1 are counted by A059167.
Sequence in context: A034125 A196013 A272366 * A075926 A368532 A065660
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 26 2019
STATUS
approved