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BII-numbers of hypertrees.
14

%I #7 Jul 27 2019 14:57:51

%S 0,1,2,4,8,16,20,32,36,48,64,128,256,260,272,276,292,304,320,512,516,

%T 532,544,548,560,576,768,784,800,1024,1040,1056,2048,2064,2068,2080,

%U 2084,2096,2112,2304,2308,2336,2560,2564,2576,2816,3072,4096,4100,4128,4608

%N BII-numbers of hypertrees.

%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, the BII-number of {{2},{1,3}} is 18.

%C 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.

%e The sequence of all hypertrees together with their BII-numbers begins:

%e 0: {}

%e 1: {{1}}

%e 2: {{2}}

%e 4: {{1,2}}

%e 8: {{3}}

%e 16: {{1,3}}

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

%e 32: {{2,3}}

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

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

%e 64: {{1,2,3}}

%e 128: {{4}}

%e 256: {{1,4}}

%e 260: {{1,2},{1,4}}

%e 272: {{1,3},{1,4}}

%e 276: {{1,2},{1,3},{1,4}}

%e 292: {{1,2},{2,3},{1,4}}

%e 304: {{1,3},{2,3},{1,4}}

%e 320: {{1,2,3},{1,4}}

%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 density[c_]:=Total[(Length[#1]-1&)/@c]-Length[Union@@c];

%t 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]]]]]]]]];

%t Select[Range[0,1000],#==0||stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&&density[bpe/@bpe[#]]==-1&]

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

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 23 2019