A326704 - OEIS

%I #11 Jun 17 2024 07:05:19

%S 0,1,2,3,4,8,9,10,11,12,16,18,20,32,33,36,48,52,64,128,129,130,131,

%T 132,136,137,138,139,140,144,146,148,160,161,164,176,180,192,256,258,

%U 260,264,266,268,272,274,276,288,292,304,308,320,512,513,516,520,521,524

%N BII-numbers of antichains of nonempty sets.

%C A binary index of n is any position of a 1 in its reversed binary expansion. 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, it follows that the BII-number of {{2},{1,3}} is 18.

%C Elements of a set-system are sometimes called edges. In an antichain of sets, no edge is a subset or superset of any other edge.

%H John Tyler Rascoe, <a href="/A326704/b326704.txt">Table of n, a(n) for n = 1..7580</a>

%H John Tyler Rascoe, <a href="/A326704/a326704.py.txt">Python Program</a>.

%e The sequence of all antichains of nonempty sets together with their BII-numbers begins:

%e    0: {}

%e    1: {{1}}

%e    2: {{2}}

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

%e    4: {{1,2}}

%e    8: {{3}}

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

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

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

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

%e   16: {{1,3}}

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

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

%e   32: {{2,3}}

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

%e   36: {{1,2},{2,3}}

%e   48: {{1,3},{2,3}}

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

%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[100],stableQ[bpe/@bpe[#],SubsetQ]&]

%o (Python) # see linked program

%Y Antichains of sets are counted by A000372.

%Y Antichains of nonempty sets are counted by A014466.

%Y MM-numbers of antichains of multisets are A316476.

%Y BII-numbers of chains of nonempty sets are A326703.

%Y Cf. A000120, A029931, A035327, A048793, A070939, A291166, A302521, A326031, A326675, A326701, A326702.

%K nonn,base

%O 1,3

%A _Gus Wiseman_, Jul 21 2019