cp's OEIS Frontend

A326752 BII-numbers of hypertrees.

%I A326752 #7 Jul 27 2019 14:57:51
%S A326752 0,1,2,4,8,16,20,32,36,48,64,128,256,260,272,276,292,304,320,512,516,
%T A326752 532,544,548,560,576,768,784,800,1024,1040,1056,2048,2064,2068,2080,
%U A326752 2084,2096,2112,2304,2308,2336,2560,2564,2576,2816,3072,4096,4100,4128,4608
%N A326752 BII-numbers of hypertrees.
%C A326752 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 A326752 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 A326752 The sequence of all hypertrees together with their BII-numbers begins:
%e A326752     0: {}
%e A326752     1: {{1}}
%e A326752     2: {{2}}
%e A326752     4: {{1,2}}
%e A326752     8: {{3}}
%e A326752    16: {{1,3}}
%e A326752    20: {{1,2},{1,3}}
%e A326752    32: {{2,3}}
%e A326752    36: {{1,2},{2,3}}
%e A326752    48: {{1,3},{2,3}}
%e A326752    64: {{1,2,3}}
%e A326752   128: {{4}}
%e A326752   256: {{1,4}}
%e A326752   260: {{1,2},{1,4}}
%e A326752   272: {{1,3},{1,4}}
%e A326752   276: {{1,2},{1,3},{1,4}}
%e A326752   292: {{1,2},{2,3},{1,4}}
%e A326752   304: {{1,3},{2,3},{1,4}}
%e A326752   320: {{1,2,3},{1,4}}
%t A326752 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326752 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326752 density[c_]:=Total[(Length[#1]-1&)/@c]-Length[Union@@c];
%t A326752 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 A326752 Select[Range[0,1000],#==0||stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&&density[bpe/@bpe[#]]==-1&]
%Y A326752 Cf. A000120, A000272, A030019 (spanning hypertrees), A035053, A048143, A048793, A052888, A070939, A134954, A275307, A326031, A326702, A326753.
%Y A326752 Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326754 (covers).
%K A326752 nonn
%O A326752 1,3
%A A326752 _Gus Wiseman_, Jul 23 2019