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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 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 September 22 19:36 EDT 2021. Contains 347608 sequences. (Running on oeis4.)