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A326753
Number of connected components of the set-system with BII-number n.
36
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
0,4
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.
LINKS
John Tyler Rascoe, Log scatterplot of a(n), n=0..32906.
FORMULA
a(A072639(n)) = n. - John Tyler Rascoe, Jul 15 2024
EXAMPLE
The set-system {{1,2},{1,4},{3}} with BII-number 268 has two connected components, so a(268) = 2.
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}]
PROG
(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
CROSSREFS
Positions of 0's and 1's are A326749.
Ranking sequences using BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).
Sequence in context: A278401 A069935 A283530 * A062093 A177457 A373125
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 23 2019
STATUS
approved