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 A327061 BII-numbers of pairwise intersecting set-systems where every two covered vertices appear together in some edge (cointersecting). 2
 0, 1, 2, 4, 5, 6, 8, 16, 17, 24, 32, 34, 40, 52, 64, 65, 66, 68, 69, 70, 72, 80, 81, 84, 85, 88, 96, 98, 100, 102, 104, 112, 116, 120, 128, 256, 257, 384, 512, 514, 640, 772, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1284, 1285, 1408, 1536, 1538 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence gives all BII-numbers (defined below) of pairwise intersecting set-systems whose dual is also pairwise intersecting. 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 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. LINKS EXAMPLE The sequence of all pairwise intersecting, cointersecting set-systems together with their BII-numbers begins: 0: {} 1: {{1}} 2: {{2}} 4: {{1,2}} 5: {{1},{1,2}} 6: {{2},{1,2}} 8: {{3}} 16: {{1,3}} 17: {{1},{1,3}} 24: {{3},{1,3}} 32: {{2,3}} 34: {{2},{2,3}} 40: {{3},{2,3}} 52: {{1,2},{1,3},{2,3}} 64: {{1,2,3}} 65: {{1},{1,2,3}} 66: {{2},{1,2,3}} 68: {{1,2},{1,2,3}} 69: {{1},{1,2},{1,2,3}} 70: {{2},{1,2},{1,2,3}} MATHEMATICA dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}]; bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1]; stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}]; Select[Range[0, 100], stableQ[bpe/@bpe[#], Intersection[#1, #2]=={}&]&&stableQ[dual[bpe/@bpe[#]], Intersection[#1, #2]=={}&]&] CROSSREFS The unlabeled multiset partition version is A319765. Equals the intersection of A326853 and A326910. The T_0 version is A326854. These set-systems are counted by A327037 (covering) and A327038 (not covering). Cf. A029931, A048793, A051185, A305843, A319774, A326031, A327039, A327040, A327041, A327057. Sequence in context: A185867 A326910 A326905 * A326913 A326703 A244008 Adjacent sequences: A327058 A327059 A327060 * A327062 A327063 A327064 KEYWORD nonn AUTHOR Gus Wiseman, Aug 18 2019 STATUS approved

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Last modified March 22 12:29 EDT 2023. Contains 361423 sequences. (Running on oeis4.)