A327105 - OEIS

%I #6 Sep 01 2019 08:40:49

%S 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 28,29,32,33,34,35,36,37,38,39,40,41,43,44,46,48,49,50,56,57,58,64,65,

%U 66,67,68,69,70,71,72,73,74,80,81,88,89,96,98,104,106,128

%N BII-numbers of set-systems with minimum 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 minimum degree 1 together with their BII-numbers begins:

%e    1: {{1}}

%e    2: {{2}}

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

%e    4: {{1,2}}

%e    5: {{1},{1,2}}

%e    6: {{2},{1,2}}

%e    8: {{3}}

%e    9: {{1},{3}}

%e   10: {{2},{3}}

%e   11: {{1},{2},{3}}

%e   12: {{1,2},{3}}

%e   13: {{1},{1,2},{3}}

%e   14: {{2},{1,2},{3}}

%e   15: {{1},{2},{1,2},{3}}

%e   16: {{1,3}}

%e   17: {{1},{1,3}}

%e   18: {{2},{1,3}}

%e   19: {{1},{2},{1,3}}

%e   20: {{1,2},{1,3}}

%e   21: {{1},{1,2},{1,3}}

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

%t Select[Range[0,100],If[#==0,0,Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]]==1&]

%Y Positions of 1's in A327103.

%Y BII-numbers for minimum degree > 1 are A327107.

%Y Graphs with minimum degree 1 are counted by A245797, with covering case A327227.

%Y Set-systems with minimum degree 1 are counted by A327228, with covering case A327229.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 26 2019