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A326754
BII-numbers of set-systems covering an initial interval of positive integers.
31
0, 1, 3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
OFFSET
1,3
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
EXAMPLE
The sequence of all covering set-systems together with their BII-numbers begins:
0: {}
1: {{1}}
3: {{1},{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
7: {{1},{2},{1,2}}
11: {{1},{2},{3}}
12: {{1,2},{3}}
13: {{1},{1,2},{3}}
14: {{2},{1,2},{3}}
15: {{1},{2},{1,2},{3}}
18: {{2},{1,3}}
19: {{1},{2},{1,3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
22: {{2},{1,2},{1,3}}
23: {{1},{2},{1,2},{1,3}}
26: {{2},{3},{1,3}}
27: {{1},{2},{3},{1,3}}
28: {{1,2},{3},{1,3}}
29: {{1},{1,2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Select[Range[0, 100], normQ[Join@@bpe/@bpe[#]]&]
PROG
(Python)
from itertools import chain, count, islice
def bin_i(n): #binary indices
return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
for n in count(0):
s = set(i for i in chain.from_iterable([bin_i(k) for k in bin_i(n)]))
y = len(s)
if sum(s) == (y*(y+1))//2:
yield n
A326754_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 20 2024
CROSSREFS
Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees).
Sequence in context: A163078 A050034 A039056 * A047562 A354270 A137922
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 23 2019
STATUS
approved