A326752 BII-numbers of hypertrees.

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 64, 128, 256, 260, 272, 276, 292, 304, 320, 512, 516, 532, 544, 548, 560, 576, 768, 784, 800, 1024, 1040, 1056, 2048, 2064, 2068, 2080, 2084, 2096, 2112, 2304, 2308, 2336, 2560, 2564, 2576, 2816, 3072, 4096, 4100, 4128, 4608

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. A hypertree is a connected antichain of nonempty sets with density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices.

Links

Table of n, a(n) for n=1..51.

Example

The sequence of all hypertrees 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}}
   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}}
  320: {{1,2,3},{1,4}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
density[c_]:=Total[(Length[#1]-1&)/@c]-Length[Union@@c];
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]]]]]]]]];
Select[Range[0, 1000], #==0||stableQ[bpe/@bpe[#], SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&&density[bpe/@bpe[#]]==-1&]

Crossrefs

Cf. A000120, A000272, A030019 (spanning hypertrees), A035053, A048143, A048793, A052888, A070939, A134954, A275307, A326031, A326702, A326753.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326754 (covers).

Sequence in context: A082003 A326750 A329561 * A093107 A125508 A067945

Adjacent sequences: A326749 A326750 A326751 * A326753 A326754 A326755

Keyword

nonn

Author

Gus Wiseman, Jul 23 2019

Status

approved