A327107 - OEIS

%I #7 Sep 01 2019 08:40:34

%S 7,25,30,31,42,45,47,51,52,53,54,55,59,60,61,62,63,75,76,77,78,79,82,

%T 83,84,85,86,87,90,91,92,93,94,95,97,99,100,101,102,103,105,107,108,

%U 109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124

%N BII-numbers of set-systems with minimum vertex-degree > 1.

%C 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.

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

%e The sequence of all set-systems with maximum degree > 1 together with their BII-numbers begins:

%e    7: {{1},{2},{1,2}}

%e   25: {{1},{3},{1,3}}

%e   30: {{2},{1,2},{3},{1,3}}

%e   31: {{1},{2},{1,2},{3},{1,3}}

%e   42: {{2},{3},{2,3}}

%e   45: {{1},{1,2},{3},{2,3}}

%e   47: {{1},{2},{1,2},{3},{2,3}}

%e   51: {{1},{2},{1,3},{2,3}}

%e   52: {{1,2},{1,3},{2,3}}

%e   53: {{1},{1,2},{1,3},{2,3}}

%e   54: {{2},{1,2},{1,3},{2,3}}

%e   55: {{1},{2},{1,2},{1,3},{2,3}}

%e   59: {{1},{2},{3},{1,3},{2,3}}

%e   60: {{1,2},{3},{1,3},{2,3}}

%e   61: {{1},{1,2},{3},{1,3},{2,3}}

%e   62: {{2},{1,2},{3},{1,3},{2,3}}

%e   63: {{1},{2},{1,2},{3},{1,3},{2,3}}

%e   75: {{1},{2},{3},{1,2,3}}

%e   76: {{1,2},{3},{1,2,3}}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[100],Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]>1&]

%Y Positions of terms > 1 in A327103.

%Y BII-numbers for minimum degree 1 are A327105.

%Y Graphs with minimum degree > 1 are counted by A059167.

%Y Cf. A000120, A029931, A048793, A058891, A070939, A245797, A326031, A326701, A326786, A327041, A327104, A327227-A327230.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 26 2019