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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

Table of n, a(n) for n=1..45.

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.)