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A327110

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

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`