OFFSET
1,3
COMMENTS
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, the BII-number of {{2},{1,3}} is 18.
Elements of a set-system are sometimes called edges. In an antichain, no edge is a subset or superset of any other edge.
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 1..6834
John Tyler Rascoe, Python program.
EXAMPLE
The sequence of all clutters together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
4: {{1,2}}
8: {{3}}
16: {{1,3}}
20: {{1,2},{1,3}}
32: {{2,3}}
36: {{1,2},{2,3}}
48: {{1,3},{2,3}}
52: {{1,2},{1,3},{2,3}}
64: {{1,2,3}}
128: {{4}}
256: {{1,4}}
260: {{1,2},{1,4}}
272: {{1,3},{1,4}}
276: {{1,2},{1,3},{1,4}}
292: {{1,2},{2,3},{1,4}}
304: {{1,3},{2,3},{1,4}}
308: {{1,2},{1,3},{2,3},{1,4}}
320: {{1,2,3},{1,4}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[0, 1000], stableQ[bpe/@bpe[#], SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&]
PROG
(Python) # see linked program
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 23 2019
STATUS
approved