So, all four of your 9/73 solutions are identical when we permit multiplication by values coprime to 73. If we start with the first one: {0,1,3,7,15,31,36,54,63} multiplication by 49 gives us {0,49,1,51,5,59,12,18,21} which sorted is the second {0,1,5,12,18,21,49,51,59} multiplication by 51 gives us the third {0,1,7,11,35,48,51,53,65} and multiplication by 33 gives us the last. {0,1,12,20,26,30,33,35,57} On Wed, Nov 11, 2020 at 1:54 PM ed pegg <ed@mathpuzzle.com> wrote:
How are you eliminating redundant sets? Adding all values and sorting mod n, then picking the first.
If I've calculated them right, the number of distinct cyclic difference sets, or the number of cyclically perfect partitions starts out 1, 2, 1, 5, 6, 4, 6, 18, 20, 6, 51, 42 {{{7,3},1}, {{13,4},2}, {{21,5},1}, {{31,6},5}, {{57,8},6}, {{73,9},4}, {{91,10},6}, {{133,12},18}, {{183,14},20}, {{273,17},6}, {{307,18},51}, {{381,20},42}}
For example, the four cyclically perfect partitions of length 9 are {{1,2,4,8,16,5,18,9,10}, {1,4,7,6,3,28,2,8,14}, {1,6,4,24,13,3,2,12,8}, {1,11,8,6,4,3,2,22,16}} All sub-arcs of a perfect partition have a unique total and cover all integer values up to the set total. These correspond to the following difference sets: {{{73,9},{0,1,3,7,15,31,36,54,63}}, {{73,9},{0,1,5,12,18,21,49,51,59}}, {{73,9},{0,1,7,11,35,48,51,53,65}}, {{73,9},{0,1,12,20,26,30,33,35,57}}}
In case anyone is interested, here are the perfect partitions. {{1,2,4}, {1,2,6,4}, {1,3,2,7}, {1,3,10,2,5}, {1,2,5,4,6,13}, {1,2,7,4,12,5}, {1,3,2,7,8,10}, {1,3,6,2,5,14}, {1,7,3,2,4,14}, {1,2,10,19,4,7,9,5}, {1,3,5,11,2,12,17,6}, {1,3,8,2,16,7,15,5}, {1,4,2,10,18,3,11,8}, {1,4,22,7,3,6,2,12}, {1,6,12,4,21,3,2,8}, {1,2,4,8,16,5,18,9,10}, {1,4,7,6,3,28,2,8,14}, {1,6,4,24,13,3,2,12,8}, {1,11,8,6,4,3,2,22,16}, {1,2,6,18,22,7,5,16,4,10}, {1,3,9,11,6,8,2,5,28,18}, {1,4,2,20,8,9,23,10,3,11}, {1,4,3,10,2,9,14,16,6,26}, {1,5,4,13,3,8,7,12,2,36}, {1,6,9,11,29,4,8,2,3,18}, {1,2,9,8,14,4,43,7,6,10,5,24}, {1,2,12,31,25,4,9,10,7,11,16,5}, {1,2,14,4,37,7,8,27,5,6,13,9}, {1,2,14,12,32,19,6,5,4,18,13,7}, {1,3,8,9,5,19,23,16,13,2,28,6}, {1,3,12,34,21,2,8,9,5,6,7,25}, {1,3,23,24,6,22,10,11,18,2,5,8}, {1,4,7,3,16,2,6,17,20,9,13,35}, {1,4,16,3,15,10,12,14,17,33,2,6}, {1,4,19,20,27,3,6,25,7,8,2,11}, {1,4,20,3,40,10,9,2,15,16,6,7}, {1,5,12,21,29,11,3,16,4,22,2,7}, {1,7,13,12,3,11,5,18,4,2,48,9}, {1,8,10,5,7,21,4,2,11,3,26,35}, {1,14,3,2,4,7,21,8,25,10,12,26}, {1,14,10,20,7,6,3,2,17,4,8,41}, {1,15,5,3,25,2,7,4,6,12,14,39}, {1,22,14,20,5,13,8,3,4,2,10,31}, {1,2,13,7,5,14,34,6,4,33,18,17,21,8}, {1,2,21,17,11,5,9,4,26,6,47,15,12,7}, {1,2,28,14,5,6,9,12,48,18,4,13,16,7}, {1,3,5,6,25,32,23,10,18,2,17,7,22,12}, {1,3,12,7,20,14,44,6,5,24,2,28,8,9}, {1,3,24,6,12,14,11,55,7,2,8,5,16,19}, {1,4,6,31,3,13,2,7,14,12,17,46,8,19}, {1,4,8,52,3,25,18,2,9,24,6,10,7,14}, {1,4,20,2,12,3,6,7,33,11,8,10,35,31}, {1,5,2,24,15,29,14,21,13,4,33,3,9,10}, {1,5,23,27,42,3,4,11,2,19,12,10,16,8}, {1,6,8,22,4,5,33,21,3,20,32,16,2,10}, {1,8,3,10,23,5,56,4,2,14,15,17,7,18}, {1,8,21,45,6,7,11,17,3,2,10,4,23,25}, {1,9,5,40,3,4,21,35,16,18,2,6,11,12}, {1,9,14,26,4,2,11,5,3,12,27,34,7,28}, {1,9,21,25,3,4,8,5,6,16,2,36,14,33}, {1,10,22,34,27,12,3,4,2,14,24,5,8,17}, {1,10,48,9,19,4,8,6,7,17,3,2,34,15}, {1,12,48,6,2,38,3,22,7,10,11,5,4,14}, {1,2,4,8,16,32,27,26,11,9,45,13,10,29,5,17,18}, {1,3,12,10,31,7,27,2,6,5,19,20,62,14,9,28,17}, {1,7,3,15,33,5,24,68,2,14,6,17,4,9,19,12,34}, {1,7,12,44,25,41,9,17,4,6,22,33,13,2,3,11,23}, {1,7,31,2,11,3,9,36,17,4,22,6,18,72,5,10,19}, {1,21,11,50,39,13,6,4,14,16,25,26,3,2,7,8,27}, {1,2,18,4,6,37,9,14,79,7,8,11,16,13,32,12,5,33}, {1,2,20,6,12,47,15,30,27,21,4,39,7,10,34,19,5,8}, {1,2,24,15,4,16,12,25,38,50,14,17,5,8,10,11,49,6}, {1,2,27,7,13,5,21,22,19,12,4,24,32,10,23,71,6,8}, {1,2,41,12,5,6,8,7,9,4,29,36,18,22,10,27,25,45}, {1,2,42,13,4,11,23,14,12,20,7,29,18,71,10,9,16,5}, {1,3,8,45,9,18,10,6,7,17,2,29,15,5,59,38,14,21}, {1,3,11,13,5,20,10,7,51,6,16,34,26,19,2,44,31,8}, {1,3,11,20,7,33,10,9,17,22,6,2,16,5,32,12,13,88}, {1,3,20,9,58,19,2,25,36,13,22,31,12,5,11,26,8,6}, {1,3,25,7,20,30,46,15,38,10,6,5,13,47,2,17,14,8}, {1,3,34,2,45,7,18,5,48,16,10,57,6,13,14,8,9,11}, {1,4,2,16,12,13,8,24,38,44,15,27,10,9,17,3,11,53}, {1,4,3,36,23,6,18,22,9,2,19,63,14,34,16,10,15,12}, {1,4,6,17,13,8,16,9,20,2,12,72,15,3,32,7,19,51}, {1,4,16,7,17,25,8,26,3,6,13,39,2,12,18,64,36,10}, {1,4,27,13,70,17,6,29,22,2,9,3,7,18,30,8,26,15}, {1,4,27,43,6,11,2,7,3,18,16,8,14,39,25,33,15,35}, {1,4,36,35,32,49,7,6,3,14,8,12,27,19,2,24,18,10}, {1,5,2,25,4,12,44,34,18,20,19,26,21,3,11,40,13,9}, {1,5,4,45,11,27,41,7,24,26,13,8,43,16,2,12,3,19}, {1,5,9,8,4,7,32,24,18,2,35,3,10,49,29,25,16,30}, {1,5,14,4,22,12,9,50,27,51,3,8,2,15,16,32,7,29}, {1,5,22,24,12,13,8,26,9,30,2,29,19,53,37,3,4,10}, {1,5,24,4,32,17,54,15,7,9,26,52,3,8,10,2,25,13}, {1,6,9,32,18,5,14,21,4,13,11,91,2,8,12,31,3,26}, {1,6,11,3,2,28,34,35,4,38,10,27,19,24,12,13,32,8}, {1,6,36,9,16,13,26,2,3,15,4,8,65,21,35,14,23,10}, {1,6,52,16,50,3,31,5,9,15,4,13,8,2,20,35,26,11}, {1,7,3,27,6,12,23,5,19,2,51,9,4,16,34,57,17,14}, {1,7,15,24,20,9,2,3,38,13,6,26,4,12,21,79,10,17}, {1,7,35,32,36,44,4,12,17,5,23,2,24,15,31,6,3,10}, {1,9,2,3,23,7,43,4,20,16,6,19,34,51,13,8,31,17}, {1,9,2,4,43,36,5,26,8,22,46,28,7,17,3,18,19,13}, {1,9,12,16,14,3,15,8,27,31,59,2,4,7,36,39,5,19}, {1,9,29,4,20,11,45,16,65,2,3,12,6,7,19,8,14,36}, {1,12,3,5,9,27,4,2,66,10,24,25,28,11,7,19,32,22}, {1,12,9,5,38,16,28,3,32,37,2,18,58,8,7,4,6,23}, {1,12,23,3,7,21,16,18,22,2,6,11,32,20,5,4,90,14}, {1,12,34,5,16,60,36,14,6,3,15,4,7,37,17,8,2,30}, {1,12,36,19,25,7,11,4,5,30,3,21,2,8,6,72,17,28}, {1,14,39,19,2,10,28,8,35,6,3,13,4,7,18,5,32,63}, {1,15,28,6,12,11,3,5,2,20,36,35,13,4,38,9,45,24}, {1,17,2,10,4,7,38,5,8,24,15,42,6,22,3,68,9,26}, {1,17,9,31,6,2,12,16,5,47,3,4,15,10,13,11,62,43}, {1,20,14,59,19,5,3,9,33,7,4,2,16,10,15,23,37,30}, {1,22,20,7,9,10,5,29,3,8,17,13,48,4,2,12,21,76}, {1,23,26,5,11,9,18,21,13,6,22,35,71,2,8,4,3,29}, {1,27,3,13,2,6,50,5,9,11,12,10,7,19,41,53,4,34}, {1,37,11,24,26,16,29,23,18,33,6,8,13,4,3,2,10,43}, {1,39,18,4,6,76,15,11,9,3,31,2,14,5,8,17,7,41}, {1,2,9,5,48,10,19,23,7,8,13,18,4,33,71,26,6,34,20,24}, {1,2,10,15,23,14,17,4,18,8,33,56,11,5,24,20,46,32,36,6}, {1,2,40,6,14,47,5,7,9,8,10,77,31,33,11,26,4,15,13,22}, {1,3,5,12,11,26,61,24,10,56,6,7,33,2,36,15,14,16,25,18}, {1,3,18,25,14,20,30,5,31,2,11,29,12,51,32,26,48,8,9,6}, {1,3,23,19,2,14,17,24,36,13,7,8,10,12,59,11,54,29,34,5}, {1,3,48,10,2,20,47,23,8,18,7,86,21,6,9,19,11,5,24,13}, {1,4,3,38,42,20,6,48,16,9,47,19,11,33,32,2,15,12,10,13}, {1,4,14,10,7,6,3,48,2,32,43,12,15,46,8,30,22,11,47,20}, {1,4,16,6,43,7,37,3,11,17,24,12,30,32,57,2,46,15,10,8}, {1,4,17,52,15,10,13,3,8,19,12,6,14,44,2,70,9,47,7,28}, {1,5,19,7,40,28,8,12,10,23,16,18,3,14,27,2,9,4,50,85}, {1,5,24,2,21,15,18,43,28,65,7,12,8,14,3,57,9,4,35,10}, {1,6,3,28,14,4,12,17,24,36,50,26,39,5,20,2,13,8,11,62}, {1,6,4,12,26,21,20,5,9,91,19,8,29,2,13,30,32,3,33,17}, {1,7,3,57,9,17,22,5,35,2,21,15,14,4,16,12,13,6,24,98}, {1,7,24,5,21,14,9,2,52,13,3,17,10,12,6,41,15,4,34,91}, {1,7,33,3,26,16,14,9,12,25,27,11,20,96,10,5,13,4,2,47}, {1,7,45,48,47,17,3,13,26,12,18,10,4,5,6,35,27,34,2,21}, {1,7,69,48,6,18,10,27,19,41,25,4,11,2,3,33,14,12,9,22}, {1,9,14,18,11,5,3,17,13,26,44,2,4,65,15,12,28,7,66,21}, {1,10,22,13,25,18,12,39,3,37,4,20,96,6,2,7,14,5,31,16}, {1,11,26,24,32,8,13,6,9,5,2,23,42,4,70,10,44,3,31,17}, {1,12,3,5,24,30,7,4,6,36,14,38,26,23,2,33,22,68,9,18}, {1,12,51,35,14,65,4,5,6,22,34,25,23,7,17,3,16,2,8,31}, {1,13,15,31,39,11,68,23,21,3,2,7,10,8,37,16,4,32,6,34}, {1,14,7,5,19,10,6,2,36,39,11,9,23,25,3,30,72,4,13,52}, {1,14,16,20,17,7,3,2,6,46,38,4,19,9,13,21,48,39,33,25}, {1,16,22,33,91,2,7,5,6,41,15,28,8,24,3,10,21,19,4,25}, {1,16,35,23,4,15,11,29,10,18,3,64,6,66,34,2,7,5,8,24}, {1,18,7,2,8,6,71,5,33,22,29,37,3,12,49,4,30,13,11,20}, {1,18,7,6,16,38,24,3,9,5,23,33,2,8,34,11,4,89,30,20}, {1,18,15,10,7,23,8,4,24,13,16,46,22,47,9,2,3,87,6,20}, {1,18,28,32,52,6,55,11,14,2,13,22,54,5,4,8,26,7,3,20}, {1,18,35,9,2,14,13,82,43,7,8,22,4,6,49,12,5,28,3,20}, {1,22,7,3,6,15,19,26,11,14,27,88,4,8,5,42,2,18,28,35}, {1,22,9,8,64,12,43,25,2,19,7,30,5,6,4,14,20,13,3,74}, {1,22,18,46,17,2,28,3,6,5,10,38,7,13,12,4,72,43,8,26}, {1,23,17,2,8,3,25,20,14,12,6,15,16,60,5,4,35,22,7,86}, {1,28,11,23,8,36,16,9,57,7,72,12,2,4,15,5,17,10,3,45}, {1,28,50,23,13,6,2,3,9,18,16,15,22,4,55,7,10,35,25,39}, {1,66,9,8,33,2,3,18,7,19,13,29,11,34,14,6,4,12,15,77}}
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