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Maximal number of 'swaps' edit

The graphs in the main article have been constructed using the following results. It is important to note that for  , the results are not optimal, but are merely lower bounds of the true result. Although the optimal value is known for   (which is 139), the corresponding row cannot be found. Therefore the best obtainable lower bound is shown for  .

row length maximal number of iterations corresponding row
1 0 1
2 1 2, 1
3 2 2, 3, 1
3, 1, 2
4 4 2, 4, 1, 3
3, 1, 4, 2
5 7 3, 1, 4, 5, 2
6 10 3, 6, 5, 1, 4, 2
4, 1, 5, 2, 6, 3
4, 1, 6, 5, 2, 3
4, 5, 6, 2, 1, 3
5, 6, 4, 1, 3, 2
7 16 3, 1, 4, 6, 7, 5, 2
4, 7, 6, 2, 1, 5, 3
8 22 6, 1, 5, 7, 8, 3, 2, 4
9 30 6, 1, 5, 9, 7, 2, 8, 3, 4
10 38 5, 9, 1, 8, 6, 2, 10, 4, 7, 3
11 51 4, 9, 11, 6, 10, 7, 8, 2, 1, 3, 5
12 65 2, 6, 1, 10, 11, 8, 12, 3, 4, 7, 9, 5
13 80 2, 9, 4, 5, 11, 12, 10, 1, 8, 13, 3, 6, 7
14 101 9, 4, 11, 3, 1, 8, 13, 6, 2, 5, 10, 14, 12, 7
15 113 2, 9, 4, 11, 7, 1, 13, 5, 3, 14, 12, 15, 8, 10, 6
7, 12, 15, 10, 13, 4, 2, 11, 1, 8, 9, 5, 14, 3, 6
16 128* 6, 13, 5, 7, 8, 3, 2, 15, 11, 16, 10, 14, 1, 9, 12, 4
17 159 2, 10, 15, 11, 7, 14, 5, 16, 6, 4, 17, 13, 1, 3, 8, 9, 12
8, 15, 17, 13, 9, 4, 6, 3, 2, 12, 16, 14, 11, 5, 10, 1, 7
18 191 6, 14, 9, 2, 15, 8, 1, 3, 4, 12, 18, 5, 10, 13, 16, 17, 11, 7
19 221 9, 4, 19, 17, 10, 1, 11, 15, 12, 8, 5, 2, 18, 13, 16, 7, 3, 14, 6
12, 1, 18, 11, 2, 3, 14, 6, 8, 16, 5, 4, 15, 10, 13, 17, 19, 7, 9
12, 1, 18, 11, 3, 14, 2, 6, 8, 16, 5, 4, 15, 10, 13, 17, 19, 7, 9
12, 15, 11, 1, 10, 17, 19, 2, 5, 8, 9, 4, 18, 13, 16, 7, 3, 14, 6
20 249 2, 8, 11, 6, 19, 17, 13, 7, 18, 3, 9, 20, 15, 1, 12, 4, 10, 14, 5, 16
23 382 14, 10, 22, 7, 8, 11, 19, 17, 15, 6, 16, 23, 20, 9, 2, 21, 3, 1, 5, 4, 12, 18, 13
24 403 2, 11, 20, 22, 9, 1, 18, 15, 14, 24, 12, 19, 8, 3, 17, 7, 5, 21, 6, 25, 10, 16, 4, 23, 13
25 466 21, 10, 18, 9, 6, 25, 1, 24, 22, 3, 2, 4, 5, 11, 8, 13, 23, 17, 12, 14, 7, 16, 15, 20, 19
29 689 8, 3, 4, 14, 11, 16, 18, 7, 21, 9, 15, 20, 22, 5, 12, 6, 10, 2, 25, 27, 17, 13, 26, 28, 1, 29, 19, 24, 23
30 706 8, 18, 11, 14, 13, 4, 3, 12, 15, 17, 10, 2, 5, 16, 22, 7, 6, 9, 21, 19, 27, 20, 28, 30, 29, 1, 26, 24, 23, 25
31 819 16, 13, 17, 14, 26, 28, 20, 30, 1, 8, 31, 27, 23, 2, 22, 10, 7, 12, 25, 6, 11, 15, 21, 4, 5, 3, 24, 18, 19, 9, 29
16, 13, 17, 14, 26, 28, 20, 30, 1, 8, 31, 27, 23, 22, 10, 2, 7, 6, 25, 3, 11, 15, 21, 4, 5, 12, 24, 18, 19, 9, 29
35 1042 12, 3, 19, 29, 23, 25, 27, 13, 8, 14, 5, 10, 9, 4, 11, 22, 21, 15, 7, 18, 2, 17, 35, 33, 30, 6, 20, 24, 16, 26, 34, 1, 28, 31, 32
37 1296 3, 18, 4, 7, 19, 16, 12, 14, 21, 15, 13, 6, 2, 20, 9, 26, 28, 5, 10, 8, 22, 27, 32, 25, 23, 30, 11, 31, 36, 1, 24, 17, 34, 37, 33, 29, 35
40 1523 25, 37, 39, 38, 29, 30, 27, 4, 3, 28, 5, 12, 6, 11, 18, 22, 14, 15, 8, 17, 21, 20, 2, 16, 13, 40, 34, 32, 7, 23, 10, 9, 26, 35, 31, 24, 19, 33, 1, 36
41 1677 4, 39, 23, 16, 18, 38, 15, 10, 28, 29, 32, 34, 30, 13, 33, 3, 5, 12, 14, 27, 31, 41, 21, 35, 11, 6, 9, 8, 17, 20, 2, 36, 26, 22, 40, 19, 1, 7, 24, 37, 25
43 2009 12, 28, 13, 21, 26, 11, 15, 22, 2, 4, 20, 18, 23, 9, 24, 7, 27, 3, 16, 14, 25, 5, 8, 6, 17, 30, 19, 10, 33, 34, 38, 42, 41, 37, 39, 32, 29, 31, 36, 43, 35, 1, 40
45 2023 8, 28, 18, 16, 23, 22, 45, 12, 5, 25, 14, 4, 2, 20, 13, 21, 43, 35, 1, 40, 7, 26, 11, 15, 6, 17, 30, 19, 10, 33, 34, 38, 42, 41, 37, 39, 32, 29, 31, 36, 44, 9, 24, 3, 27
47 2492 13, 8, 12, 22, 24, 3, 17, 9, 23, 26, 38, 30, 7, 5, 28, 25, 20, 15, 21, 14, 18, 4, 16, 2, 41, 1, 36, 43, 46, 35, 34, 27, 19, 6, 11, 42, 32, 10, 33, 31, 29, 47, 37, 45, 39, 40, 44
50 2620 19, 31, 24, 27, 25, 16, 39, 40, 49, 47, 2, 23, 26, 29, 22, 8, 15, 7, 21, 28, 20, 9, 5, 36, 33, 37, 35, 38, 11, 14, 13, 17, 18, 6, 10, 4, 3, 12, 32, 30, 44, 34, 42, 41, 50, 45, 43, 1, 46, 48
53 3185 12, 8, 14, 19, 9, 2, 22, 15, 17, 13, 16, 3, 6, 5, 11, 4, 7, 21, 10, 26, 24, 20, 27, 18, 28, 34, 23, 25, 32, 33, 29, 41, 30, 31, 40, 38, 44, 39, 49, 43, 35, 37, 36, 51, 42, 53, 1, 52, 50, 45, 46, 48, 47
55 3215 16, 13, 45, 43, 46, 19, 14, 10, 26, 3, 2, 29, 42, 35, 27, 24, 5, 6, 21, 31, 8, 15, 9, 4, 28, 7, 23, 18, 11, 12, 20, 25, 22, 36, 32, 37, 33, 30, 34, 38, 52, 41, 55, 47, 49, 39, 40, 44, 17, 54, 53, 48, 51, 1, 50
59 4393 13, 58, 56, 1, 57, 59, 55, 49, 18, 46, 51, 41, 23, 25, 39, 32, 5, 20, 14, 33, 30, 47, 40, 27, 8, 37, 9, 48, 36, 16, 43, 54, 50, 44, 22, 3, 26, 34, 31, 2, 11, 24, 15, 21, 29, 35, 52, 42, 19, 6, 28, 38, 4, 12, 45, 10, 17, 7, 53
60 4717 35, 37, 59, 57, 53, 46, 54, 36, 24, 33, 20, 22, 43, 31, 15, 27, 25, 9, 6, 14, 48, 55, 17, 38, 34, 21, 11, 28, 23, 2, 12, 41, 30, 4, 42, 45, 8, 40, 50, 47, 52, 19, 5, 29, 39, 10, 16, 26, 7, 44, 3, 32, 18, 13, 49, 60, 51, 1, 56, 58
61 4858 4, 54, 7, 25, 28, 17, 35, 42, 15, 31, 46, 51, 1, 56, 58, 21, 47, 2, 37, 9, 3, 12, 49, 43, 19, 50, 36, 30, 8, 38, 23, 33, 13, 24, 20, 41, 16, 27, 45, 39, 11, 6, 52, 14, 29, 34, 10, 55, 22, 40, 44, 60, 32, 48, 18, 5, 59, 26, 61, 57, 53
65 4955 6, 19, 15, 12, 27, 2, 25, 33, 20, 22, 16, 21, 23, 11, 10, 28, 14, 34, 37, 38, 36, 4, 18, 8, 39, 29, 5, 40, 9, 13, 7, 24, 17, 3, 26, 30, 32, 44, 46, 42, 47, 45, 35, 43, 31, 52, 41, 53, 48, 49, 57, 50, 51, 58, 59, 64, 56, 54, 55, 1, 63, 60, 65, 62, 61
67 5695 11, 43, 39, 10, 16, 18, 48, 7, 29, 14, 13, 38, 6, 60, 58, 55, 49, 52, 51, 15, 31, 24, 34, 19, 3, 17, 25, 36, 4, 12, 20, 57, 50, 42, 32, 28, 21, 9, 23, 33, 30, 45, 26, 27, 22, 41, 8, 2, 40, 5, 35, 44, 46, 37, 47, 64, 54, 61, 53, 67, 59, 66, 1, 63, 62, 56, 65
70 5926 34, 5, 38, 33, 8, 32, 6, 41, 12, 2, 28, 4, 25, 22, 46, 37, 47, 45, 43, 24, 36, 31, 18, 30, 48, 40, 26, 23, 39, 57, 50, 17, 29, 21, 11, 14, 60, 20, 16, 44, 27, 10, 35, 7, 19, 13, 9, 64, 54, 61, 53, 69, 59, 66, 70, 63, 62, 56, 1, 68, 58, 55, 49, 52, 51, 15, 3, 42, 67, 65
71 6847 5, 34, 6, 36, 31, 33, 18, 19, 24, 27, 4, 37, 43, 50, 56, 9, 32, 17, 25, 26, 30, 2, 15, 28, 13, 46, 54, 47, 8, 20, 3, 16, 12, 7, 10, 14, 21, 11, 42, 38, 35, 23, 22, 48, 39, 44, 29, 40, 57, 41, 55, 62, 58, 53, 49, 59, 51, 52, 45, 70, 65, 67, 60, 63, 64, 69, 61, 71, 66, 1, 68
73 7212 20, 24, 7, 25, 40, 26, 39, 28, 19, 21, 18, 44, 30, 5, 46, 3, 10, 33, 22, 4, 8, 38, 14, 15, 12, 61, 58, 35, 41, 32, 11, 45, 34, 17, 36, 42, 49, 47, 52, 43, 2, 27, 16, 23, 13, 62, 1, 71, 64, 56, 6, 57, 48, 37, 9, 63, 31, 50, 60, 55, 59, 53, 68, 66, 73, 67, 69, 72, 54, 65, 51, 29, 70
75 7240 2, 43, 67, 73, 66, 68, 53, 59, 55, 12, 61, 60, 50, 31, 63, 9, 37, 48, 57, 6, 56, 64, 71, 1, 62, 13, 23, 58, 35, 41, 32, 11, 24, 75, 19, 28, 39, 26, 7, 21, 18, 27, 16, 52, 47, 49, 42, 33, 22, 4, 3, 46, 5, 25, 40, 17, 30, 44, 45, 34, 10, 38, 8, 20, 36, 15, 14, 69, 72, 54, 65, 51, 29, 70, 74
79 8781 4, 17, 7, 14, 13, 15, 16, 28, 20, 36, 5, 25, 30, 9, 11, 2, 3, 31, 26, 6, 18, 32, 19, 34, 27, 41, 38, 35, 39, 33, 37, 45, 46, 51, 47, 44, 22, 12, 23, 21, 29, 10, 24, 8, 40, 43, 42, 53, 61, 68, 60, 57, 56, 58, 48, 52, 55, 50, 49, 54, 59, 75, 67, 79, 66, 1, 76, 64, 62, 71, 65, 63, 74, 78, 77, 72, 70, 69, 73
80 8782 80, 73, 69, 70, 72, 77, 78, 74, 63, 65, 71, 62, 64, 76, 1, 66, 79, 67, 75, 59, 54, 49, 50, 55, 52, 48, 58, 56, 57, 60, 68, 61, 53, 42, 43, 40, 8, 24, 10, 29, 21, 23, 12, 22, 44, 47, 51, 46, 45, 37, 33, 39, 35, 38, 41, 27, 34, 19, 32, 18, 6, 26, 31, 3, 2, 11, 9, 30, 25, 5, 36, 20, 28, 16, 15, 13, 14, 7, 17, 4
83 9938 8, 52, 18, 19, 25, 62, 34, 49, 43, 27, 22, 66, 40, 17, 51, 41, 55, 24, 42, 16, 7, 5, 36, 9, 35, 50, 30, 63, 69, 32, 12, 47, 78, 83, 45, 54, 64, 3, 10, 59, 48, 13, 68, 75, 82, 80, 72, 76, 29, 31, 2, 26, 4, 37, 33, 53, 46, 44, 21, 38, 74, 70, 11, 65, 15, 20, 58, 56, 23, 39, 57, 6, 28, 14, 67, 71, 61, 60, 77, 73, 1, 79, 81
85 9972 15, 69, 63, 18, 52, 5, 12, 47, 78, 83, 64, 54, 10, 59, 48, 20, 58, 56, 51, 17, 40, 66, 65, 85, 45, 77, 60, 61, 71, 67, 76, 72, 50, 35, 9, 25, 62, 34, 73, 1, 79, 81, 27, 22, 46, 44, 6, 57, 13, 68, 16, 43, 8, 53, 33, 37, 4, 39, 23, 3, 7, 14, 2, 31, 49, 11, 70, 74, 38, 21, 28, 26, 30,36, 32, 19, 29, 41, 55, 24, 42, 84, 80, 82, 75
89 11840 29, 13, 45, 59, 65, 56, 6, 33, 36, 22, 51, 9, 35, 19, 12, 62, 39, 21, 10, 40, 53, 63, 54, 32, 66, 64, 49, 55, 17, 26, 2, 46, 7, 42, 8, 52, 43, 34, 23, 50, 57, 14, 28, 69, 18, 48, 20, 31, 30, 67, 5, 4, 61, 58, 16, 47, 60, 41, 25, 75, 74, 38, 3, 72, 70, 24, 68, 27, 11, 73, 76, 15, 37, 44, 78, 71, 88, 80, 84, 77, 86, 83, 81, 82, 87, 79, 89, 1, 85
90 11846 15, 62, 39, 21, 10, 40, 52, 37, 27, 68, 67, 5, 4, 69, 18, 76, 73, 11, 43, 34, 23, 50, 57, 25, 41, 60, 56, 6, 14, 28, 65, 42, 7, 3, 72, 70, 24, 44, 78, 71, 88, 80, 84, 77, 86, 83, 81, 82, 87, 79, 89, 1, 85, 90, 59, 75, 74, 38, 8, 46, 2, 26, 29, 13, 33, 20, 31, 30, 61, 58, 16, 47, 19, 12, 48, 45, 35, 53, 63, 54, 32, 66, 64, 49, 55, 17, 9, 51, 22, 36
95 12925 9, 30, 43, 48, 82, 80, 72, 61, 18, 86, 90, 87, 79, 52, 26, 31, 64, 44, 10, 78, 3, 6, 71, 16, 83, 66, 58, 70, 38, 81, 4, 27, 50, 25, 21, 49, 12, 17, 53, 34, 59, 33, 67, 45, 60, 35, 54, 76, 69, 75, 47, 37, 46, 77, 40, 41, 19, 62, 57, 55, 39, 84, 73, 23, 68, 8, 65, 42, 13, 24, 32, 14, 20, 5, 29, 2, 56, 28, 15, 36, 51, 22, 63, 7, 91, 74, 11, 1, 94, 88, 89, 95, 85, 92, 93
97 15008 42, 6, 39, 10, 12, 29, 20, 64, 60, 68, 65, 59, 61, 66, 57, 58, 56, 78, 70, 74, 67, 71, 54, 55, 53, 62, 40, 52, 35, 45, 32, 25, 37, 26, 2, 77, 80, 82, 72, 79, 73, 13, 5, 27, 3, 36, 4, 44, 47, 41, 22, 18, 33, 11, 48, 51, 14, 21, 24, 46, 31, 63, 49, 16, 43, 9, 50, 34, 30, 19, 38, 7, 8, 23, 15, 17, 28, 83, 76, 86, 69, 75, 89, 81, 97, 85, 92, 93, 96, 87, 90, 91, 94, 1, 84, 88, 95
42, 3, 39, 10, 12, 29, 20, 64, 60, 68, 65, 59, 61, 66, 57, 58, 56, 78, 70, 74, 67, 71, 54, 55, 53, 62, 40, 52, 35, 45, 32, 25, 37, 26, 2, 77, 80, 82, 72, 79, 73, 13, 6, 27, 5, 36, 4, 44, 47, 41, 22, 18, 33, 11, 48, 51, 14, 21, 24, 46, 31, 63, 49, 16, 43, 9, 50, 34, 30, 19, 38, 7, 8, 23, 15, 17, 28, 83, 76, 86, 69, 75, 89, 81, 97, 85, 92, 93, 96, 87, 90, 91, 94, 1, 84, 88, 95
100 15045 10, 100, 98, 95, 88, 84, 1, 68, 60, 64, 6, 42, 80, 77, 83, 76, 86, 69, 75, 89, 81, 97, 85, 92, 93, 96, 52, 35, 13, 73, 28, 99, 38, 19, 78, 43, 9, 15, 3, 27, 46, 31, 63, 5, 39, 2, 17, 65, 59, 61, 66, 57, 79, 72, 82, 23, 7, 11, 33, 51, 14, 34, 30, 8, 20, 70, 74, 29, 12, 16, 49, 67, 71, 54, 55, 53, 62, 40, 56, 58, 41, 22, 18, 45, 32, 44, 47, 50, 21, 25, 37, 26, 36, 4, 48, 24, 87, 90, 91, 94
105 15961 5, 7, 2, 18, 6, 3, 15, 4, 10, 13, 12, 14, 9, 8, 38, 11, 26, 16, 21, 24, 31, 29, 33, 42, 43, 62, 51, 41, 83, 71, 39, 47, 56, 55, 44, 53, 49, 54, 40, 46, 17, 23, 30, 22, 19, 20, 25, 34, 37, 27, 35, 28, 32, 36, 48, 52, 45, 50, 65, 67, 60, 61, 57, 59, 58, 68, 63, 72, 64, 66, 69, 77, 76, 89, 84, 81, 79, 70, 78, 74, 73, 91, 86, 87, 75, 80, 82, 96, 95, 92, 97, 85, 103, 101, 93, 90, 88, 102, 94, 104, 99, 98, 105, 100, 1
110 17897 5, 7, 2, 18, 6, 3, 15, 4, 10, 13, 12, 14, 9, 8, 38, 11, 26, 16, 21, 24, 31, 29, 33, 42, 43, 62, 51, 41, 83, 71, 39, 47, 56, 55, 44, 53, 49, 54, 40, 46, 17, 23, 30, 22, 19, 20, 25, 34, 37, 27, 35, 28, 32, 36, 48, 52, 45, 50, 65, 67, 60, 61, 57, 59, 58, 68, 63, 72, 64, 66, 69, 77, 76, 85, 84, 81, 79, 70, 78, 74, 73, 90, 86, 87, 75, 80, 82, 89, 92, 88, 100, 94, 98, 91, 106, 97, 95, 96, 103, 101, 110, 1, 93, 105, 107, 99, 102, 109, 104, 108
115 20326 8, 9, 14, 12, 13, 10, 4, 7, 5, 3, 2, 18, 6, 15, 11, 26, 16, 21, 17, 23, 24, 30, 22, 19, 20, 25, 34, 37, 27, 35, 28, 32, 36, 31, 29, 33, 42, 43, 62, 51, 41, 38, 40, 46, 39, 47, 56, 55, 44, 53, 49, 54, 48, 52, 45, 50, 65, 67, 60, 61, 57, 59, 58, 68, 63, 72, 64, 66, 69, 83, 71, 77, 76, 90, 84, 81, 79, 70, 78, 74, 73, 95, 86, 87, 75, 80, 82, 91, 92, 85, 93, 97, 89, 96, 88, 98, 94, 100, 109, 102, 106, 99, 103, 105, 111, 104, 114, 112, 1, 110, 108, 101, 115, 107, 113
120 23506 7, 51, 34, 73, 16, 100, 5, 60, 67, 12, 8, 105, 109, 63, 79, 28, 95, 52, 75, 71, 41, 80, 58, 116, 119, 1, 120, 110, 117, 111, 106, 99, 38, 30, 43, 72, 74, 66, 64, 26, 77, 69, 54, 92, 103, 13, 31, 90, 102, 25, 4, 17, 42, 87, 70, 57, 36, 33, 20, 22, 2, 97, 93, 88, 96, 53, 35, 101, 32, 23, 49, 44, 48, 94, 86, 81, 85, 59, 3, 83, 10, 65, 21, 29, 98, 19, 24, 68, 14, 9, 50, 89, 18, 84, 76, 39, 62, 27, 78, 82, 108, 118, 104, 91, 37, 112, 46, 40, 6, 15, 11, 61, 47, 45, 56, 55, 114, 115, 107, 113
125 26572 2, 6, 11, 10, 12, 8, 4, 9, 7, 3, 5, 18, 17, 13, 74, 70, 90, 84, 28, 48, 44, 47, 59, 57, 53, 51, 49, 46, 42, 45, 52, 50, 60, 55, 68, 56, 58, 64, 63, 62, 73, 69, 72, 76, 67, 61, 66, 71, 85, 86, 80, 91, 89, 93, 88, 87, 97, 95, 103, 94, 98, 92, 101, 96, 107, 100, 99, 102, 110, 106, 104, 105, 124, 122, 117, 113, 108, 116, 111, 109, 118, 112, 114, 119, 82, 79, 81, 78, 65, 75, 77, 83, 14, 22, 23, 20, 29, 34, 26, 25, 27, 21, 38, 33, 30, 40, 31, 43, 41, 39, 35, 32, 36, 37, 54, 15, 16, 19, 24, 115, 125, 120, 1, 121, 123
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