OFFSET
1,3
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. 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, it follows that the BII-number of {{2},{1,3}} is 18.
Elements of a set-system are sometimes called edges. In a chain of sets, every edge is a subset or superset of every other edge.
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 1..4860
EXAMPLE
The sequence of all chains of nonempty sets together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
8: {{3}}
16: {{1,3}}
17: {{1},{1,3}}
24: {{3},{1,3}}
32: {{2,3}}
34: {{2},{2,3}}
40: {{3},{2,3}}
64: {{1,2,3}}
65: {{1},{1,2,3}}
66: {{2},{1,2,3}}
68: {{1,2},{1,2,3}}
69: {{1},{1,2},{1,2,3}}
70: {{2},{1,2},{1,2,3}}
72: {{3},{1,2,3}}
80: {{1,3},{1,2,3}}
81: {{1},{1,3},{1,2,3}}
88: {{3},{1,3},{1,2,3}}
96: {{2,3},{1,2,3}}
98: {{2},{2,3},{1,2,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[0, 100], stableQ[bpe/@bpe[#], !SubsetQ[#1, #2]&&!SubsetQ[#2, #1]&]&]
PROG
(Python)
from itertools import chain, count, combinations, islice
from sympy.combinatorics.subsets import ksubsets
def subsets(x):
for i in range(1, len(x)):
for j in ksubsets(x, i):
yield(list(j))
def a_gen(): #generator of terms
yield 0
for n in count(1):
t, v, j = [[]], [], 0
for i in chain.from_iterable(combinations(range(1, n+1), r) for r in range(n+1)):
if n in i:
t[j].append([list(i)])
while n:
t.append([])
for i in t[j]:
if len(i[-1]) > 1:
for k in list(subsets(i[-1])):
t[j+1].append(i.copy()+[k])
if len(t[j+1]) < 1:
break
j += 1
for j in chain.from_iterable(t):
v.append(sum(2**(sum(2**(m-1) for m in k)-1) for k in j))
yield from sorted(v)
A326703_list = list(islice(a_gen(), 55)) # John Tyler Rascoe, Jun 07 2024
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Jul 21 2019
STATUS
approved