cp's OEIS Frontend

A327107 BII-numbers of set-systems with minimum vertex-degree > 1.

%I A327107 #7 Sep 01 2019 08:40:34
%S A327107 7,25,30,31,42,45,47,51,52,53,54,55,59,60,61,62,63,75,76,77,78,79,82,
%T A327107 83,84,85,86,87,90,91,92,93,94,95,97,99,100,101,102,103,105,107,108,
%U A327107 109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124
%N A327107 BII-numbers of set-systems with minimum vertex-degree > 1.
%C A327107 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 A327107 In a set-system, the degree of a vertex is the number of edges containing it.
%e A327107 The sequence of all set-systems with maximum degree > 1 together with their BII-numbers begins:
%e A327107    7: {{1},{2},{1,2}}
%e A327107   25: {{1},{3},{1,3}}
%e A327107   30: {{2},{1,2},{3},{1,3}}
%e A327107   31: {{1},{2},{1,2},{3},{1,3}}
%e A327107   42: {{2},{3},{2,3}}
%e A327107   45: {{1},{1,2},{3},{2,3}}
%e A327107   47: {{1},{2},{1,2},{3},{2,3}}
%e A327107   51: {{1},{2},{1,3},{2,3}}
%e A327107   52: {{1,2},{1,3},{2,3}}
%e A327107   53: {{1},{1,2},{1,3},{2,3}}
%e A327107   54: {{2},{1,2},{1,3},{2,3}}
%e A327107   55: {{1},{2},{1,2},{1,3},{2,3}}
%e A327107   59: {{1},{2},{3},{1,3},{2,3}}
%e A327107   60: {{1,2},{3},{1,3},{2,3}}
%e A327107   61: {{1},{1,2},{3},{1,3},{2,3}}
%e A327107   62: {{2},{1,2},{3},{1,3},{2,3}}
%e A327107   63: {{1},{2},{1,2},{3},{1,3},{2,3}}
%e A327107   75: {{1},{2},{3},{1,2,3}}
%e A327107   76: {{1,2},{3},{1,2,3}}
%t A327107 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327107 Select[Range[100],Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]>1&]
%Y A327107 Positions of terms > 1 in A327103.
%Y A327107 BII-numbers for minimum degree 1 are A327105.
%Y A327107 Graphs with minimum degree > 1 are counted by A059167.
%Y A327107 Cf. A000120, A029931, A048793, A058891, A070939, A245797, A326031, A326701, A326786, A327041, A327104, A327227-A327230.
%K A327107 nonn
%O A327107 1,1
%A A327107 _Gus Wiseman_, Aug 26 2019