cp's OEIS Frontend

A309314 BII-numbers of hyperforests.

%I A309314 #17 Jul 27 2019 14:57:51
%S A309314 0,1,2,3,4,8,9,10,11,12,16,18,20,32,33,36,48,64,128,129,130,131,132,
%T A309314 136,137,138,139,140,144,146,148,160,161,164,176,192,256,258,260,264,
%U A309314 266,268,272,274,276,288,292,304,320,512,513,516,520,521,524,528,532
%N A309314 BII-numbers of hyperforests.
%C A309314 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 A309314 Elements of a set-system are sometimes called edges. In an antichain, no edge is a subset or superset of any other edge. A hyperforest is an antichain of nonempty sets whose connected components are hypertrees, meaning they have density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices.
%e A309314 The sequence of all hyperforests together with their BII-numbers begins:
%e A309314     0: {}
%e A309314     1: {{1}}
%e A309314     2: {{2}}
%e A309314     3: {{1},{2}}
%e A309314     4: {{1,2}}
%e A309314     8: {{3}}
%e A309314     9: {{1},{3}}
%e A309314    10: {{2},{3}}
%e A309314    11: {{1},{2},{3}}
%e A309314    12: {{1,2},{3}}
%e A309314    16: {{1,3}}
%e A309314    18: {{2},{1,3}}
%e A309314    20: {{1,2},{1,3}}
%e A309314    32: {{2,3}}
%e A309314    33: {{1},{2,3}}
%e A309314    36: {{1,2},{2,3}}
%e A309314    48: {{1,3},{2,3}}
%e A309314    64: {{1,2,3}}
%e A309314   128: {{4}}
%e A309314   129: {{1},{4}}
%e A309314   130: {{2},{4}}
%e A309314   131: {{1},{2},{4}}
%e A309314   132: {{1,2},{4}}
%e A309314   136: {{3},{4}}
%e A309314   137: {{1},{3},{4}}
%Y A309314 Cf. A000120, A030019, A035053, A048143, A048793, A052888, A070939, A134954, A275307, A326031, A326702, A326753.
%Y A309314 Other BII-numbers: A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).
%K A309314 nonn
%O A309314 1,3
%A A309314 _Gus Wiseman_, Jul 23 2019