cp's OEIS Frontend

A326753 Number of connected components of the set-system with BII-number n.

%I A326753 #25 Jul 17 2024 08:47:59
%S A326753 0,1,1,2,1,1,1,1,1,2,2,3,2,2,2,2,1,1,2,2,1,1,1,1,1,1,2,2,1,1,1,1,1,2,
%T A326753 1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A326753 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A326753 Number of connected components of the set-system with BII-number n.
%C A326753 A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. 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.
%H A326753 John Tyler Rascoe, <a href="/A326753/b326753.txt">Table of n, a(n) for n = 0..10000</a>
%H A326753 John Tyler Rascoe, <a href="/A326753/a326753_3.png">Log scatterplot of a(n)</a>, n=0..32906.
%F A326753 a(A072639(n)) = n. - _John Tyler Rascoe_, Jul 15 2024
%e A326753 The set-system {{1,2},{1,4},{3}} with BII-number 268 has two connected components, so a(268) = 2.
%t A326753 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326753 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 A326753 Table[Length[csm[bpe/@bpe[n]]],{n,0,100}]
%o A326753 (Python)
%o A326753 from sympy.utilities.iterables import connected_components
%o A326753 def bin_i(n): #binary indices
%o A326753     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A326753 def A326753(n):
%o A326753     E,a = [],[bin_i(k) for k in bin_i(n)]
%o A326753     m = len(a)
%o A326753     for i in range(m):
%o A326753         for j in a[i]:
%o A326753             for k in range(m):
%o A326753                 if j in a[k]:
%o A326753                     E.append((i,k))
%o A326753     return(len(connected_components((list(range(m)),E)))) # _John Tyler Rascoe_, Jul 16 2024
%Y A326753 Positions of 0's and 1's are A326749.
%Y A326753 Cf. A000120, A001187, A029931, A048143, A048793, A070939, A072639, A304716, A305078, A305079 (same for MM-numbers), A323818, A326031, A326702.
%Y A326753 Ranking sequences using BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).
%K A326753 nonn,base
%O A326753 0,4
%A A326753 _Gus Wiseman_, Jul 23 2019