A326979 - OEIS

%I #5 Aug 13 2019 13:20:31

%S 0,1,2,3,7,8,9,10,11,15,25,27,30,31,42,43,45,47,51,52,53,54,55,59,60,

%T 61,62,63,75,79,91,94,95,107,109,111,115,116,117,118,119,123,124,125,

%U 126,127,128,129,130,131,135,136,137,138,139,143,153,155,158,159

%N BII-numbers of T_1 set-systems.

%C A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_1 condition means that the dual is a (strict) antichain, meaning that none of its edges is a subset of any other.

%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 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.

%e The sequence of all T_1 set-systems together with their BII-numbers begins:

%e    0: {}

%e    1: {{1}}

%e    2: {{2}}

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

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

%e    8: {{3}}

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

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

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

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

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

%e   27: {{1},{2},{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   43: {{1},{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}}

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

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Select[Range[0,100],UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[dual[bpe/@bpe[#]],SubsetQ]&]

%Y BII-numbers of T_0 set-systems are A326947.

%Y T_1 set-systems are counted by A326965, A326961 (covering), A326972 (unlabeled), and A326974 (unlabeled covering).

%Y BII-numbers of set-systems whose dual is a weak antichain are A326966.

%Y Cf. A059201, A059523, A319639, A326970, A326973, A326976, A326977.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 13 2019