A329560 - OEIS

%I #7 Nov 29 2019 01:39:23

%S 0,3,9,10,11,12,18,33,52,129,130,131,132,136,137,138,139,140,144,146,

%T 148,160,161,164,176,180,192,258,264,266,268,274,288,292,304,308,513,

%U 520,521,524,528,532,545,560,564,772,776,780,784,788,800,804,816,820,832

%N BII-numbers of antichains of sets with empty intersection.

%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 of positive integers) 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 A set-system is an antichain if no edge is a proper subset of any other.

%C Empty intersection means there is no vertex in common to all the edges

%e The sequence of terms together with their binary expansions and corresponding set-systems begins:

%e     0:          0 ~ {}

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

%e     9:       1001 ~ {{1},{3}}

%e    10:       1010 ~ {{2},{3}}

%e    11:       1011 ~ {{1},{2},{3}}

%e    12:       1100 ~ {{1,2},{3}}

%e    18:      10010 ~ {{2},{1,3}}

%e    33:     100001 ~ {{1},{2,3}}

%e    52:     110100 ~ {{1,2},{1,3},{2,3}}

%e   129:   10000001 ~ {{1},{4}}

%e   130:   10000010 ~ {{2},{4}}

%e   131:   10000011 ~ {{1},{2},{4}}

%e   132:   10000100 ~ {{1,2},{4}}

%e   136:   10001000 ~ {{3},{4}}

%e   137:   10001001 ~ {{1},{3},{4}}

%e   138:   10001010 ~ {{2},{3},{4}}

%e   139:   10001011 ~ {{2},{3},{4}}

%e   140:   10001100 ~ {{1,2},{3},{4}}

%e   144:   10010000 ~ {{1,3},{4}}

%e   146:   10010010 ~ {{2},{1,3},{4}}

%e   148:   10010100 ~ {{1,2},{1,3},{4}}

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

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

%t Select[Range[0,100],#==0||Intersection@@bpe/@bpe[#]=={}&&stableQ[bpe/@bpe[#],SubsetQ]&]

%Y Intersection of A326911 and A326704.

%Y BII-numbers of intersecting set-systems with empty intersecting are A326912.

%Y Cf. A000120, A048793, A070939, A326031, A326701, A328671, A329561, A329626, A329628, A329661.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 28 2019