cp's OEIS Frontend

A327105 BII-numbers of set-systems with minimum degree 1.

%I A327105 #6 Sep 01 2019 08:40:49
%S A327105 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,
%T A327105 28,29,32,33,34,35,36,37,38,39,40,41,43,44,46,48,49,50,56,57,58,64,65,
%U A327105 66,67,68,69,70,71,72,73,74,80,81,88,89,96,98,104,106,128
%N A327105 BII-numbers of set-systems with minimum degree 1.
%C A327105 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 A327105 In a set-system, the degree of a vertex is the number of edges containing it.
%e A327105 The sequence of all set-systems with minimum degree 1 together with their BII-numbers begins:
%e A327105    1: {{1}}
%e A327105    2: {{2}}
%e A327105    3: {{1},{2}}
%e A327105    4: {{1,2}}
%e A327105    5: {{1},{1,2}}
%e A327105    6: {{2},{1,2}}
%e A327105    8: {{3}}
%e A327105    9: {{1},{3}}
%e A327105   10: {{2},{3}}
%e A327105   11: {{1},{2},{3}}
%e A327105   12: {{1,2},{3}}
%e A327105   13: {{1},{1,2},{3}}
%e A327105   14: {{2},{1,2},{3}}
%e A327105   15: {{1},{2},{1,2},{3}}
%e A327105   16: {{1,3}}
%e A327105   17: {{1},{1,3}}
%e A327105   18: {{2},{1,3}}
%e A327105   19: {{1},{2},{1,3}}
%e A327105   20: {{1,2},{1,3}}
%e A327105   21: {{1},{1,2},{1,3}}
%t A327105 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327105 Select[Range[0,100],If[#==0,0,Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]]==1&]
%Y A327105 Positions of 1's in A327103.
%Y A327105 BII-numbers for minimum degree > 1 are A327107.
%Y A327105 Graphs with minimum degree 1 are counted by A245797, with covering case A327227.
%Y A327105 Set-systems with minimum degree 1 are counted by A327228, with covering case A327229.
%Y A327105 Cf. A000120, A029931, A048793, A058891, A070939, A326031, A326701, A326786, A327041, A327104, A327230.
%K A327105 nonn
%O A327105 1,2
%A A327105 _Gus Wiseman_, Aug 26 2019