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

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Author

Gus Wiseman, Jul 23 2019

Keywords

• nonn
• base

Comments

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.

Examples

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

Links

• John Tyler Rascoe, Table of n, a(n) for n = 0..10000
• John Tyler Rascoe, Log scatterplot of a(n), n=0..32906.

Crossrefs

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

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

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

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
Table[Length[csm[bpe/@bpe[n]]],{n,0,100}]

Python

from sympy.utilities.iterables import connected_components
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def A326753(n):
    E,a = [],[bin_i(k) for k in bin_i(n)]
    m = len(a)
    for i in range(m):
        for j in a[i]:
            for k in range(m):
                if j in a[k]:
                    E.append((i,k))
    return(len(connected_components((list(range(m)),E)))) # John Tyler Rascoe, Jul 16 2024

Formula

a(A072639(n)) = n. - John Tyler Rascoe, Jul 15 2024