List of the entries of 3X3 matrices with non-negative elements <=n such that the 9! matrices formed by chosing all possible permutations of the given set of matrix elements produces the maximum possible number of different determinants. URL of this file: http://www.randomwalk.de/sequences/a099834.txt Author: Hugo Pfoertner http://www.pfoertner.org/ Change history: Feb 11 2005 "10078"-result fro n=97 Jan 17 2005 "10078"-result for n=98 Jan 13 2005 "10078"-result for n=98 Jan 03 2005 2 "10078"-results for n=95 Dec 13 2004 Solution with 10080 different determinants found for n=99 Dec 10 2004 More results with 10078 different determinants for n=99,98,... Nov 29 2004 Check multiple solutions: n=31...34 Nov 25 2004 Check multiple solutions: n=30 Nov 24 2004 Check multiple solutions: n=26...29 Nov 22 2004 n=44,45,46 new, check for multiple solutions: n=21...25 Nov 19 2004 Extended n=41,42,43 Nov 17 2004 Initial version Multi- Number of different determinants Maximum determinant ple | that can be formed sols. | Largest element from optimal set. | | | Elements of optimal matrix (OEIS A099815) 1 5 1 1 1 1 1 1 0 0 0 2 1 15 2 2 2 1 1 1 0 0 0 9 <.. 2 15 2 2 2 1 1 1 1 0 0 7 <.. multiple solutions 3 15 2 2 2 2 1 1 1 0 0 8 <.. 1 53 3 3 3 2 2 1 1 0 0 28 1 109 4 4 4 3 3 2 1 1 0 62 1 209 5 5 5 4 3 2 1 1 0 124 <.. multiple solutions 2 209 5 5 5 4 4 3 2 1 0 123 <.. 1 351 6 6 5 5 4 3 1 1 0 202 1 573 7 7 6 6 5 4 2 1 0 331 1 811 8 8 7 6 5 3 2 1 0 456 1 1193 9 9 8 7 6 5 2 1 0 724 1 1509 10 9 9 8 7 5 2 1 0 937 1 1971 11 11 10 9 7 6 2 1 0 1391 1 2501 12 11 10 9 8 7 3 1 0 1526 1 3183 13 12 11 10 9 7 2 1 0 2084 1 3769 14 13 12 11 10 5 3 1 0 2424 1 4511 15 14 13 12 11 5 2 1 0 3107 1 5025 16 15 14 13 11 5 2 1 0 3771 1 5641 17 16 15 14 13 6 3 1 0 4694 1 6165 18 17 16 15 13 7 3 1 0 5704 1 6600 19 18 17 16 14 9 3 1 0 7119 1 6964 20 19 18 17 16 7 3 1 0 8062 Assumption: All matrix elements are distinct (i.e. nothing like for n=11) 1 7354 21 20 19 18 17 8 3 1 0 9632 1 7696 22 21 20 19 17 8 3 1 0 10987 1 7960 23 22 21 19 17 8 3 1 0 12332 1 8110 24 23 22 21 19 9 4 1 0 14506 1 8404 25 24 23 22 21 9 4 1 0 16626 1 8606 26 25 24 23 22 11 5 1 0 19296 1 8704 27 26 25 24 23 20 11 1 0 22492 1 8846 28 27 26 25 23 21 18 1 0 21669 <--- first decrease 1 8962 29 28 27 25 24 11 9 1 0 25179 of max determinant 1 9125 30 29 28 27 25 23 19 1 0 27430 1 9210 31 30 29 28 27 26 19 1 0 32044 1 9284 32 30 29 28 27 25 19 1 0 32555 1 9362 33 32 31 29 25 14 5 1 0 39916 1 9420 34 33 32 29 27 13 10 1 0 41032 Assumption: Matrix contains elements 0, 1. n=35 and n=36 will be checked without this assumption, but the required computer time (~= 2*1 week CPU on an 1.5GHz Intel Itanium 2) will not be available before Christmas holidays 2004) 1 9494 35 33 32 31 27 11 5 1 0 43516 1 9510 36 35 34 33 31 11 8 1 0 50016 1 9606 37 36 34 33 29 15 13 1 0 51969 1 9634 38 37 35 33 31 20 8 1 0 62923 1 9670 39 38 37 35 32 31 22 1 0 64415 1 9684 40 38 37 34 33 27 25 1 0 60559 <--- decrease 1 9716 41 40 39 37 29 18 6 1 0 77902 <.. 2 9716 41 40 39 37 31 18 7 1 0 78452 <.. multiple solutions 1 9750 42 40 39 38 37 33 23 1 0 81463 1 9786 43 42 40 39 37 35 26 1 0 86114 1 9814 44 43 39 37 34 15 6 1 0 87686 1 9822 45 44 41 38 37 16 5 1 0 98768 1 9852 46 45 43 41 37 13 9 1 0 102035 <.. 2 9852 46 45 44 43 41 38 29 1 0 109649 <.. multiple solutions ... ... examples just falling short by one pair of determinants from the maximum: 1 10078 95 93 89 88 67 53 41 1 0 899175 2 10078 95 93 89 87 77 50 27 1 0 993954 1 10078 97 96 93 87 83 74 47 1 0 1074604 2 10078 97 95 93 90 82 73 51 1 0 1052804 3 10078 97 92 89 87 79 66 12 1 0 1170098 1 10078 98 97 95 83 61 54 43 1 0 949965 2 10078 98 97 91 89 87 71 53 1 0 1066037 3 10078 98 96 95 89 86 53 35 1 0 1114941 4 10078 98 96 95 83 67 43 23 1 0 1032496 5 10078 98 96 94 93 73 65 29 1 0 1141439 6 10078 98 95 89 79 75 27 6 1 0 967022 7 10078 98 94 86 79 71 70 45 1 0 907288 Found by exhaustive brute force search: 10080 99 97 89 87 61 54 20 1 0 1039208 The previous result makes the following sequence obsolete: Found by massive random search (OEIS A098072): 10080 100 91 88 87 82 43 17 1 0 1035772