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 A326979 BII-numbers of T_1 set-systems. 17
 0, 1, 2, 3, 7, 8, 9, 10, 11, 15, 25, 27, 30, 31, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 79, 91, 94, 95, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 135, 136, 137, 138, 139, 143, 153, 155, 158, 159 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A set-system is a finite set of finite nonempty sets. 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}}. The T_1 condition means that the dual is a (strict) antichain, meaning that none of its edges is a subset of any other. 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 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. LINKS Table of n, a(n) for n=1..60. EXAMPLE The sequence of all T_1 set-systems together with their BII-numbers begins: 0: {} 1: {{1}} 2: {{2}} 3: {{1},{2}} 7: {{1},{2},{1,2}} 8: {{3}} 9: {{1},{3}} 10: {{2},{3}} 11: {{1},{2},{3}} 15: {{1},{2},{1,2},{3}} 25: {{1},{3},{1,3}} 27: {{1},{2},{3},{1,3}} 30: {{2},{1,2},{3},{1,3}} 31: {{1},{2},{1,2},{3},{1,3}} 42: {{2},{3},{2,3}} 43: {{1},{2},{3},{2,3}} 45: {{1},{1,2},{3},{2,3}} 47: {{1},{2},{1,2},{3},{2,3}} 51: {{1},{2},{1,3},{2,3}} 52: {{1,2},{1,3},{2,3}} MATHEMATICA bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1]; dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}]; stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}]; Select[Range[0, 100], UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[dual[bpe/@bpe[#]], SubsetQ]&] CROSSREFS BII-numbers of T_0 set-systems are A326947. T_1 set-systems are counted by A326965, A326961 (covering), A326972 (unlabeled), and A326974 (unlabeled covering). BII-numbers of set-systems whose dual is a weak antichain are A326966. Cf. A059201, A059523, A319639, A326970, A326973, A326976, A326977. Sequence in context: A299497 A047534 A353651 * A326874 A118374 A344624 Adjacent sequences: A326976 A326977 A326978 * A326980 A326981 A326982 KEYWORD nonn AUTHOR Gus Wiseman, Aug 13 2019 STATUS approved

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Last modified September 29 02:45 EDT 2023. Contains 365749 sequences. (Running on oeis4.)