cp's OEIS Frontend

A327110 BII-numbers of set-systems with spanning edge-connectivity 3.

%I A327110 #4 Oct 04 2019 23:29:43
%S A327110 116,117,118,119,124,125,126,127,1796,1797,1798,1799,1904,1905,1906,
%T A327110 1907,1908,1909,1910,1911,1912,1913,1914,1915,1916,1917,1918,1919,
%U A327110 1924,1925,1926,1927,2032,2033,2034,2035,2036,2037,2038,2039,2040,2041,2042,2043,2044
%N A327110 BII-numbers of set-systems with spanning edge-connectivity 3.
%C A327110 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 A327110 The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.
%e A327110 The sequence of all set-systems with spanning edge-connectivity 3 together with their BII-numbers begins:
%e A327110    116: {{1,2},{1,3},{2,3},{1,2,3}}
%e A327110    117: {{1},{1,2},{1,3},{2,3},{1,2,3}}
%e A327110    118: {{2},{1,2},{1,3},{2,3},{1,2,3}}
%e A327110    119: {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
%e A327110    124: {{1,2},{3},{1,3},{2,3},{1,2,3}}
%e A327110    125: {{1},{1,2},{3},{1,3},{2,3},{1,2,3}}
%e A327110    126: {{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
%e A327110    127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
%e A327110   1796: {{1,2},{1,4},{2,4},{1,2,4}}
%e A327110   1797: {{1},{1,2},{1,4},{2,4},{1,2,4}}
%e A327110   1798: {{2},{1,2},{1,4},{2,4},{1,2,4}}
%e A327110   1799: {{1},{2},{1,2},{1,4},{2,4},{1,2,4}}
%e A327110   1904: {{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
%e A327110   1905: {{1},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
%e A327110   1906: {{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
%e A327110   1907: {{1},{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
%e A327110   1908: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
%e A327110   1909: {{1},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
%e A327110   1910: {{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
%e A327110   1911: {{1},{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}
%t A327110 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A327110 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A327110 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t A327110 Select[Range[1000],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==3&]
%Y A327110 Positions of 3's in A327144.
%Y A327110 BII-numbers for spanning edge-connectivity 2 are A327108.
%Y A327110 BII-numbers for spanning edge-connectivity >= 2 are A327109.
%Y A327110 BII-numbers for spanning edge-connectivity 1 are A327111.
%Y A327110 Cf. A001187, A095983, A322395, A326749, A326753, A326787, A327069, A327071, A327103, A327130, A327146, A327147.
%K A327110 nonn
%O A327110 1,1
%A A327110 _Gus Wiseman_, Oct 03 2019