cp's OEIS Frontend

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

%I A326966 #6 Aug 13 2019 13:19:52
%S A326966 0,1,2,3,4,7,8,9,10,11,12,15,16,18,25,27,30,31,32,33,42,43,45,47,51,
%T A326966 52,53,54,55,59,60,61,62,63,64,75,76,79,82,91,94,95,97,107,109,111,
%U A326966 115,116,117,118,119,123,124,125,126,127,128,129,130,131,132,135
%N A326966 BII-numbers of set-systems whose dual is a weak antichain.
%C A326966 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
%C A326966 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 A326966 The sequence of all set-systems whose dual is a weak antichain together with their BII-numbers begins:
%e A326966    0: {}
%e A326966    1: {{1}}
%e A326966    2: {{2}}
%e A326966    3: {{1},{2}}
%e A326966    4: {{1,2}}
%e A326966    7: {{1},{2},{1,2}}
%e A326966    8: {{3}}
%e A326966    9: {{1},{3}}
%e A326966   10: {{2},{3}}
%e A326966   11: {{1},{2},{3}}
%e A326966   12: {{1,2},{3}}
%e A326966   15: {{1},{2},{1,2},{3}}
%e A326966   16: {{1,3}}
%e A326966   18: {{2},{1,3}}
%e A326966   25: {{1},{3},{1,3}}
%e A326966   27: {{1},{2},{3},{1,3}}
%e A326966   30: {{2},{1,2},{3},{1,3}}
%e A326966   31: {{1},{2},{1,2},{3},{1,3}}
%e A326966   32: {{2,3}}
%e A326966   33: {{1},{2,3}}
%t A326966 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326966 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326966 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326966 Select[Range[0,100],stableQ[dual[bpe/@bpe[#]],SubsetQ]&]
%Y A326966 Set-systems whose dual is a weak antichain are counted by A326968, with covering case A326970, unlabeled version A326971, and unlabeled covering version A326973.
%Y A326966 BII-numbers of set-systems whose dual is strict (T_0) are A326947.
%Y A326966 BII-numbers of set-systems whose dual is a (strict) antichain (T_1) are A326979.
%Y A326966 Cf. A059523, A319639, A326961, A326965, A326969, A326974, A326975, A326978.
%K A326966 nonn
%O A326966 1,3
%A A326966 _Gus Wiseman_, Aug 13 2019