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 A340135 Number of pairs of independent nontrivial subsets of a finite set composed of n elements. 0
 0, 0, 0, 0, 24, 0, 720, 0, 7000, 15120, 126000, 0, 1777776, 0, 23543520, 55855800, 274565720, 0, 5337775872, 0, 63026049424, 117920013120, 995265791520, 0, 15265486117744, 14283091977000, 216344919117600, 240142901941800, 2854493961432480, 0, 55689696384165720 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS A subset X of a set S is called a trivial subset, if it is equal to the empty set or to the full set S. The subsets A and B of a finite set S are called independent, if #A/#S * #B/#S = #(A \intersect B)/#S). LINKS Table of n, a(n) for n=0..30. Jochen Ziegenbalg, Independent subsets EXAMPLE For n=4 and S={1,2,3,4} the a(4)=24 pairs of independent nontrivial subsets of S are {{1, 2}, {1, 3}}, {{1, 2}, {1, 4}}, {{1, 2}, {2, 3}}, {{1, 2}, {2, 4}}, {{1, 3}, {1, 2}}, {{1, 3}, {1, 4}}, {{1, 3}, {2, 3}}, {{1, 3}, {3, 4}}, {{1, 4}, {1, 2}}, {{1, 4}, {1, 3}}, {{1, 4}, {2, 4}}, {{1, 4}, {3, 4}}, {{2, 3}, {1, 2}}, {{2, 3}, {1, 3}}, {{2, 3}, {2, 4}}, {{2, 3}, {3, 4}}, {{2, 4}, {1, 2}}, {{2, 4}, {1, 4}}, {{2, 4}, {2, 3}}, {{2, 4}, {3, 4}}, {{3, 4}, {1, 3}}, {{3, 4}, {1, 4}}, {{3, 4}, {2, 3}}, {{3, 4}, {2, 4}} Tables: n all independent independent independent proper nontrivial subsets subsets subsets (see A121312) (see A158345) a(n) 0 1 0 0 1 4 1 0 2 12 5 0 3 28 13 0 4 84 53 24 5 124 61 0 6 972 845 720 7 508 253 0 8 8020 7509 7000 9 17164 16141 15120 10 130092 128045 126000 11 8188 4093 0 12 1794156 1785965 1777776 13 32764 16381 0 14 23609052 23576285 23543520 15 55986868 55921333 55855800 16 274827860 274696789 274565720 17 524284 262141 0 18 5338824444 5338300157 5337775872 19 2097148 1048573 0 20 63030243724 63028146573 63026049424 21 117928401724 117924207421 117920013120 22 995282568732 995274180125 995265791520 23 33554428 16777213 0 24 15265553226604 15265519672173 15265486117744 25 14283226194724 14283159085861 14283091977000 26 216345187553052 216345053335325 216344919117600 27 240143438812708 240143170377253 240142901941800 28 2854495035174300 2854494498303389 2854493961432480 29 2147483644 1073741821 0 30 55689700679133012 55689698531649365 55689696384165720 PROG (Maxima) /* version 1 */ pairs_independent_nontrivial_subsets(n) := block([a, b, d, s : 0 ], for a:1 thru n-1 do for d:1 thru a do ( b : n*d / a, if integerp(b) and b

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Last modified May 29 15:14 EDT 2023. Contains 363042 sequences. (Running on oeis4.)