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A326703 BII-numbers of chains of nonempty sets.

%I A326703 #21 Jun 08 2024 01:47:56
%S A326703 0,1,2,4,5,6,8,16,17,24,32,34,40,64,65,66,68,69,70,72,80,81,88,96,98,
%T A326703 104,128,256,257,384,512,514,640,1024,1025,1026,1028,1029,1030,1152,
%U A326703 1280,1281,1408,1536,1538,1664,2048,2056,2176,4096,4097,4104,4112,4113,4120
%N A326703 BII-numbers of chains of nonempty sets.
%C A326703 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.
%C A326703 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.
%H A326703 John Tyler Rascoe, <a href="/A326703/b326703.txt">Table of n, a(n) for n = 1..4860</a>
%e A326703 The sequence of all chains of nonempty sets together with their BII-numbers begins:
%e A326703     0: {}
%e A326703     1: {{1}}
%e A326703     2: {{2}}
%e A326703     4: {{1,2}}
%e A326703     5: {{1},{1,2}}
%e A326703     6: {{2},{1,2}}
%e A326703     8: {{3}}
%e A326703    16: {{1,3}}
%e A326703    17: {{1},{1,3}}
%e A326703    24: {{3},{1,3}}
%e A326703    32: {{2,3}}
%e A326703    34: {{2},{2,3}}
%e A326703    40: {{3},{2,3}}
%e A326703    64: {{1,2,3}}
%e A326703    65: {{1},{1,2,3}}
%e A326703    66: {{2},{1,2,3}}
%e A326703    68: {{1,2},{1,2,3}}
%e A326703    69: {{1},{1,2},{1,2,3}}
%e A326703    70: {{2},{1,2},{1,2,3}}
%e A326703    72: {{3},{1,2,3}}
%e A326703    80: {{1,3},{1,2,3}}
%e A326703    81: {{1},{1,3},{1,2,3}}
%e A326703    88: {{3},{1,3},{1,2,3}}
%e A326703    96: {{2,3},{1,2,3}}
%e A326703    98: {{2},{2,3},{1,2,3}}
%t A326703 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326703 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326703 Select[Range[0,100],stableQ[bpe/@bpe[#],!SubsetQ[#1,#2]&&!SubsetQ[#2,#1]&]&]
%o A326703 (Python)
%o A326703 from itertools import chain, count, combinations, islice
%o A326703 from sympy.combinatorics.subsets import ksubsets
%o A326703 def subsets(x):
%o A326703     for i in range(1,len(x)):
%o A326703         for j in ksubsets(x,i):
%o A326703             yield(list(j))
%o A326703 def a_gen(): #generator of terms
%o A326703     yield 0
%o A326703     for n in count(1):
%o A326703         t,v,j = [[]],[],0
%o A326703         for i in chain.from_iterable(combinations(range(1, n+1), r) for r in range(n+1)):
%o A326703             if n in i:
%o A326703                 t[j].append([list(i)])
%o A326703         while n:
%o A326703             t.append([])
%o A326703             for i in t[j]:
%o A326703                 if len(i[-1]) > 1:
%o A326703                     for k in list(subsets(i[-1])):
%o A326703                         t[j+1].append(i.copy()+[k])
%o A326703             if len(t[j+1]) < 1:
%o A326703                 break
%o A326703             j += 1
%o A326703         for j in chain.from_iterable(t):
%o A326703             v.append(sum(2**(sum(2**(m-1) for m in k)-1) for k in j))
%o A326703         yield from sorted(v)
%o A326703 A326703_list = list(islice(a_gen(), 55)) # _John Tyler Rascoe_, Jun 07 2024
%Y A326703 Chains of nonempty sets are counted by A000629.
%Y A326703 MM-numbers of chains of multisets are A318991.
%Y A326703 BII-numbers of antichains of nonempty sets are A326704.
%Y A326703 Cf. A000120, A014466, A029931, A035327, A048793, A070939, A326031, A326701, A326702.
%K A326703 nonn,base
%O A326703 1,3
%A A326703 _Gus Wiseman_, Jul 21 2019