cp's OEIS Frontend

A327234 Smallest BII-number of a set-system with cut-connectivity n.

%I A327234 #7 Sep 09 2019 12:03:56
%S A327234 0,1,4,52,2868
%N A327234 Smallest BII-number of a set-system with cut-connectivity n.
%C A327234 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 set-system (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.
%C A327234 We define the cut-connectivity (A326786) of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853), this is the same as vertex-connectivity (A327051).
%C A327234 Conjecture: a(n > 1) = A327373(n) = the BII-number of K_n.
%e A327234 The sequence of terms together with their corresponding set-systems:
%e A327234      0: {}
%e A327234      1: {{1}}
%e A327234      4: {{1,2}}
%e A327234     52: {{1,2},{1,3},{2,3}}
%e A327234   2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
%Y A327234 The same for spanning edge-connectivity is A327147.
%Y A327234 The cut-connectivity of the set-system with BII-number n is A326786(n).
%Y A327234 Cf. A000120, A002450, A029931, A048793, A070939, A259862, A326031, A327082, A327098, A327125, A327126, A327127, A327373.
%K A327234 nonn,more
%O A327234 0,3
%A A327234 _Gus Wiseman_, Sep 03 2019