cp's OEIS Frontend

A326979 BII-numbers of T_1 set-systems.

%I A326979 #5 Aug 13 2019 13:20:31
%S A326979 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 A326979 61,62,63,75,79,91,94,95,107,109,111,115,116,117,118,119,123,124,125,
%U A326979 126,127,128,129,130,131,135,136,137,138,139,143,153,155,158,159
%N A326979 BII-numbers of T_1 set-systems.
%C A326979 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 A326979 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 A326979 The sequence of all T_1 set-systems together with their BII-numbers begins:
%e A326979    0: {}
%e A326979    1: {{1}}
%e A326979    2: {{2}}
%e A326979    3: {{1},{2}}
%e A326979    7: {{1},{2},{1,2}}
%e A326979    8: {{3}}
%e A326979    9: {{1},{3}}
%e A326979   10: {{2},{3}}
%e A326979   11: {{1},{2},{3}}
%e A326979   15: {{1},{2},{1,2},{3}}
%e A326979   25: {{1},{3},{1,3}}
%e A326979   27: {{1},{2},{3},{1,3}}
%e A326979   30: {{2},{1,2},{3},{1,3}}
%e A326979   31: {{1},{2},{1,2},{3},{1,3}}
%e A326979   42: {{2},{3},{2,3}}
%e A326979   43: {{1},{2},{3},{2,3}}
%e A326979   45: {{1},{1,2},{3},{2,3}}
%e A326979   47: {{1},{2},{1,2},{3},{2,3}}
%e A326979   51: {{1},{2},{1,3},{2,3}}
%e A326979   52: {{1,2},{1,3},{2,3}}
%t A326979 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326979 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326979 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326979 Select[Range[0,100],UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[dual[bpe/@bpe[#]],SubsetQ]&]
%Y A326979 BII-numbers of T_0 set-systems are A326947.
%Y A326979 T_1 set-systems are counted by A326965, A326961 (covering), A326972 (unlabeled), and A326974 (unlabeled covering).
%Y A326979 BII-numbers of set-systems whose dual is a weak antichain are A326966.
%Y A326979 Cf. A059201, A059523, A319639, A326970, A326973, A326976, A326977.
%K A326979 nonn
%O A326979 1,3
%A A326979 _Gus Wiseman_, Aug 13 2019