cp's OEIS Frontend

A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

%I A367917 #13 Dec 28 2023 11:33:59
%S A367917 0,1,2,3,5,6,8,9,10,11,13,14,17,19,21,22,24,26,28,34,35,37,38,40,41,
%T A367917 44,49,50,52,56,67,69,70,73,74,76,81,82,84,88,97,98,100,104,112,128,
%U A367917 129,130,131,133,134,136,137,138,139,141,142,145,147,149,150,152
%N A367917 BII-numbers of set-systems with the same number of edges as covered vertices.
%C A367917 A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. 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.
%e A367917 The terms together with the corresponding set-systems begin:
%e A367917    0: {}
%e A367917    1: {{1}}
%e A367917    2: {{2}}
%e A367917    3: {{1},{2}}
%e A367917    5: {{1},{1,2}}
%e A367917    6: {{2},{1,2}}
%e A367917    8: {{3}}
%e A367917    9: {{1},{3}}
%e A367917   10: {{2},{3}}
%e A367917   11: {{1},{2},{3}}
%e A367917   13: {{1},{1,2},{3}}
%e A367917   14: {{2},{1,2},{3}}
%e A367917   17: {{1},{1,3}}
%e A367917   19: {{1},{2},{1,3}}
%e A367917   21: {{1},{1,2},{1,3}}
%e A367917   22: {{2},{1,2},{1,3}}
%e A367917   24: {{3},{1,3}}
%e A367917   26: {{2},{3},{1,3}}
%e A367917   28: {{1,2},{3},{1,3}}
%e A367917   34: {{2},{2,3}}
%e A367917   35: {{1},{2},{2,3}}
%e A367917   37: {{1},{1,2},{2,3}}
%t A367917 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
%t A367917 Select[Range[0,100], Length[bpe[#]]==Length[Union@@bpe/@bpe[#]]&]
%Y A367917 These set-systems are counted by A054780 and A367916, A368186.
%Y A367917 Graphs of this type are A367862, covering A367863, unlabeled A006649.
%Y A367917 A003465 counts set-systems covering {1..n}, unlabeled A055621.
%Y A367917 A048793 lists binary indices, length A000120, sum A029931.
%Y A367917 A058891 counts set-systems, connected A323818, unlabeled A000612.
%Y A367917 A070939 gives length of binary expansion.
%Y A367917 A136556 counts set-systems on {1..n} with n edges.
%Y A367917 Cf. A057500, A059201, A072639, A096111, A116508, A309326, A326031, A326702, A326753, A326754, A367770, A367902, A367905.
%K A367917 nonn
%O A367917 1,3
%A A367917 _Gus Wiseman_, Dec 12 2023