login
A326750
BII-numbers of clutters (connected antichains of nonempty sets).
22
0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 52, 64, 128, 256, 260, 272, 276, 292, 304, 308, 320, 512, 516, 532, 544, 548, 560, 564, 576, 768, 772, 784, 788, 800, 804, 816, 820, 832, 1024, 1040, 1056, 1072, 1088, 2048, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304
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, Python program.
FORMULA
Intersection of A326749 and A326704.
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
The number of clutters spanning n vertices is A048143(n).
Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326751 (blobs), A326752 (hypertrees), A326754 (covers).
Sequence in context: A109713 A241582 A082003 * A329561 A326752 A093107
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 23 2019
STATUS
approved