A326753 - OEIS

%I #25 Jul 17 2024 08:47:59

%S 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 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 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

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

%C 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 John Tyler Rascoe, <a href="/A326753/b326753.txt">Table of n, a(n) for n = 0..10000</a>

%H John Tyler Rascoe, <a href="/A326753/a326753_3.png">Log scatterplot of a(n)</a>, n=0..32906.

%F a(A072639(n)) = n. - _John Tyler Rascoe_, Jul 15 2024

%e The set-system {{1,2},{1,4},{3}} with BII-number 268 has two connected components, so a(268) = 2.

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%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 Table[Length[csm[bpe/@bpe[n]]],{n,0,100}]

%o (Python)

%o from sympy.utilities.iterables import connected_components

%o def bin_i(n): #binary indices

%o     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])

%o def A326753(n):

%o     E,a = [],[bin_i(k) for k in bin_i(n)]

%o     m = len(a)

%o     for i in range(m):

%o         for j in a[i]:

%o             for k in range(m):

%o                 if j in a[k]:

%o                     E.append((i,k))

%o     return(len(connected_components((list(range(m)),E)))) # _John Tyler Rascoe_, Jul 16 2024

%Y Positions of 0's and 1's are A326749.

%Y Cf. A000120, A001187, A029931, A048143, A048793, A070939, A072639, A304716, A305078, A305079 (same for MM-numbers), A323818, A326031, A326702.

%Y Ranking sequences using BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

%K nonn,base

%O 0,4

%A _Gus Wiseman_, Jul 23 2019