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 A327110 BII-numbers of set-systems with spanning edge-connectivity 3. 1
 116, 117, 118, 119, 124, 125, 126, 127, 1796, 1797, 1798, 1799, 1904, 1905, 1906, 1907, 1908, 1909, 1910, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1918, 1919, 1924, 1925, 1926, 1927, 2032, 2033, 2034, 2035, 2036, 2037, 2038, 2039, 2040, 2041, 2042, 2043, 2044 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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. LINKS EXAMPLE The sequence of all set-systems with spanning edge-connectivity 3 together with their BII-numbers begins:    116: {{1,2},{1,3},{2,3},{1,2,3}}    117: {{1},{1,2},{1,3},{2,3},{1,2,3}}    118: {{2},{1,2},{1,3},{2,3},{1,2,3}}    119: {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}    124: {{1,2},{3},{1,3},{2,3},{1,2,3}}    125: {{1},{1,2},{3},{1,3},{2,3},{1,2,3}}    126: {{2},{1,2},{3},{1,3},{2,3},{1,2,3}}    127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}   1796: {{1,2},{1,4},{2,4},{1,2,4}}   1797: {{1},{1,2},{1,4},{2,4},{1,2,4}}   1798: {{2},{1,2},{1,4},{2,4},{1,2,4}}   1799: {{1},{2},{1,2},{1,4},{2,4},{1,2,4}}   1904: {{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}   1905: {{1},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}   1906: {{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}   1907: {{1},{2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}   1908: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}   1909: {{1},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}   1910: {{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}}   1911: {{1},{2},{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4},{1,2,4}} MATHEMATICA bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1]; 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]]]]]]]]]; spanEdgeConn[vts_, eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds], Union@@#!=vts||Length[csm[#]]!=1&]; Select[Range[1000], spanEdgeConn[Union@@bpe/@bpe[#], bpe/@bpe[#]]==3&] CROSSREFS Positions of 3's in A327144. BII-numbers for spanning edge-connectivity 2 are A327108. BII-numbers for spanning edge-connectivity >= 2 are A327109. BII-numbers for spanning edge-connectivity 1 are A327111. Cf. A001187, A095983, A322395, A326749, A326753, A326787, A327069, A327071, A327103, A327130, A327146, A327147. Sequence in context: A279451 A056101 A051116 * A255925 A095623 A257197 Adjacent sequences:  A327107 A327108 A327109 * A327111 A327112 A327113 KEYWORD nonn AUTHOR Gus Wiseman, Oct 03 2019 STATUS approved

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Last modified April 21 14:42 EDT 2021. Contains 343154 sequences. (Running on oeis4.)