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Problem Statement for "LadderPermutation"

Problem Statement

You have n distinct integers between 1 and n, inclusive. A permutation of these integers is called an (m,k)-ladder permutation if its longest increasing subsequence has length m and its longest decreasing subsequence has length k. A subsequence is a sequence created by removing zero or more elements from an original sequence. The relative order of the remaining elements must be preserved. For example, {1, 3} is a subsequence of {1, 2, 3}, but {3, 2} is not. An increasing sequence is one in which each element is greater than the previous element, and a decreasing sequence is one in which each element is less than the previous element.

You are given ints n, m and k. Return a int[] containing the (m,k)-ladder permutation of size n. If there are multiple possibilities, return the one that comes first lexicographically. If there is no such permutation, return an empty int[]. Sequence A comes before sequence B lexicographically if A contains a lower value at the first position where they differ.

Definition

Class:
LadderPermutation
Method:
createLadder
Parameters:
int, int, int
Returns:
int[]
Method signature:
int[] createLadder(int n, int m, int k)
(be sure your method is public)

Constraints

  • n will be between 1 and 50, inclusive.
  • m will be between 1 and n, inclusive.
  • k will be between 1 and n, inclusive.

Examples

  1. 4

    2

    2

    Returns: {2, 1, 4, 3 }

    In this case, all longest increasing subsequences have length 2 (for example, {1, 3}), and all longest decreasing subsequences have length 2 (for example, {2, 1}).

  2. 3

    2

    2

    Returns: {1, 3, 2 }

  3. 2

    1

    1

    Returns: { }

    In this case, the two numbers will always form an increasing or decreasing sequence of length 2. There is no permutation where the longest increasing/decreasing subsequence only has length 1.

  4. 6

    3

    2

    Returns: {2, 1, 4, 3, 6, 5 }

  5. 6

    2

    3

    Returns: {3, 2, 1, 6, 5, 4 }

  6. 7

    4

    4

    Returns: {1, 2, 3, 7, 6, 5, 4 }

  7. 6

    4

    2

    Returns: {1, 2, 4, 3, 6, 5 }

  8. 6

    4

    3

    Returns: {1, 2, 3, 6, 5, 4 }

  9. 6

    3

    3

    Returns: {1, 3, 2, 6, 5, 4 }

  10. 6

    4

    3

    Returns: {1, 2, 3, 6, 5, 4 }

  11. 6

    4

    4

    Returns: { }

  12. 7

    3

    3

    Returns: {1, 4, 3, 2, 7, 6, 5 }

  13. 9

    3

    3

    Returns: {3, 2, 1, 6, 5, 4, 9, 8, 7 }

  14. 9

    4

    2

    Returns: { }

  15. 9

    4

    3

    Returns: {1, 3, 2, 6, 5, 4, 9, 8, 7 }

  16. 9

    4

    4

    Returns: {1, 2, 5, 4, 3, 9, 8, 7, 6 }

  17. 9

    5

    5

    Returns: {1, 2, 3, 4, 9, 8, 7, 6, 5 }

  18. 9

    4

    5

    Returns: {1, 2, 4, 3, 9, 8, 7, 6, 5 }

  19. 9

    5

    4

    Returns: {1, 2, 3, 5, 4, 9, 8, 7, 6 }

  20. 9

    5

    5

    Returns: {1, 2, 3, 4, 9, 8, 7, 6, 5 }

  21. 9

    5

    6

    Returns: { }

  22. 9

    6

    5

    Returns: { }

  23. 6

    1

    5

    Returns: { }

  24. 6

    1

    6

    Returns: {6, 5, 4, 3, 2, 1 }

  25. 6

    2

    5

    Returns: {1, 6, 5, 4, 3, 2 }

  26. 6

    6

    1

    Returns: {1, 2, 3, 4, 5, 6 }

  27. 6

    5

    1

    Returns: { }

  28. 6

    5

    2

    Returns: {1, 2, 3, 4, 6, 5 }

  29. 1

    1

    1

    Returns: {1 }

  30. 2

    2

    2

    Returns: { }

  31. 50

    8

    8

    Returns: {1, 2, 10, 9, 8, 7, 6, 5, 4, 3, 18, 17, 16, 15, 14, 13, 12, 11, 26, 25, 24, 23, 22, 21, 20, 19, 34, 33, 32, 31, 30, 29, 28, 27, 42, 41, 40, 39, 38, 37, 36, 35, 50, 49, 48, 47, 46, 45, 44, 43 }

  32. 50

    8

    7

    Returns: {1, 8, 7, 6, 5, 4, 3, 2, 15, 14, 13, 12, 11, 10, 9, 22, 21, 20, 19, 18, 17, 16, 29, 28, 27, 26, 25, 24, 23, 36, 35, 34, 33, 32, 31, 30, 43, 42, 41, 40, 39, 38, 37, 50, 49, 48, 47, 46, 45, 44 }

  33. 50

    7

    8

    Returns: {2, 1, 10, 9, 8, 7, 6, 5, 4, 3, 18, 17, 16, 15, 14, 13, 12, 11, 26, 25, 24, 23, 22, 21, 20, 19, 34, 33, 32, 31, 30, 29, 28, 27, 42, 41, 40, 39, 38, 37, 36, 35, 50, 49, 48, 47, 46, 45, 44, 43 }

  34. 50

    1

    50

    Returns: {50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 }

  35. 50

    50

    1

    Returns: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 }

  36. 1

    1

    1

    Returns: {1 }

  37. 11

    4

    8

    Returns: {1, 2, 3, 11, 10, 9, 8, 7, 6, 5, 4 }

  38. 16

    3

    6

    Returns: {4, 3, 2, 1, 10, 9, 8, 7, 6, 5, 16, 15, 14, 13, 12, 11 }

  39. 22

    2

    11

    Returns: {11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12 }

  40. 4

    4

    1

    Returns: {1, 2, 3, 4 }

  41. 15

    14

    6

    Returns: { }

  42. 39

    13

    28

    Returns: { }

  43. 43

    31

    14

    Returns: { }

  44. 9

    3

    5

    Returns: {1, 4, 3, 2, 9, 8, 7, 6, 5 }

  45. 13

    11

    4

    Returns: { }

  46. 25

    4

    22

    Returns: {1, 2, 3, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4 }

  47. 15

    12

    15

    Returns: { }

  48. 25

    23

    21

    Returns: { }

  49. 2

    1

    1

    Returns: { }

  50. 26

    1

    16

    Returns: { }

  51. 1

    1

    1

    Returns: {1 }

  52. 39

    28

    22

    Returns: { }

  53. 11

    8

    7

    Returns: { }

  54. 48

    31

    48

    Returns: { }

  55. 13

    9

    4

    Returns: {1, 2, 3, 4, 5, 6, 7, 9, 8, 13, 12, 11, 10 }

  56. 5

    4

    3

    Returns: { }

  57. 45

    23

    26

    Returns: { }

  58. 48

    33

    17

    Returns: { }

  59. 1

    1

    1

    Returns: {1 }

  60. 39

    30

    8

    Returns: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31, 30, 29, 39, 38, 37, 36, 35, 34, 33, 32 }

  61. 8

    8

    7

    Returns: { }

  62. 30

    26

    21

    Returns: { }

  63. 15

    2

    10

    Returns: {5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6 }

  64. 15

    14

    3

    Returns: { }

  65. 14

    8

    14

    Returns: { }

  66. 9

    2

    3

    Returns: { }

  67. 9

    6

    5

    Returns: { }

  68. 15

    3

    10

    Returns: {1, 5, 4, 3, 2, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6 }

  69. 37

    28

    3

    Returns: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 24, 28, 27, 26, 31, 30, 29, 34, 33, 32, 37, 36, 35 }

  70. 4

    1

    1

    Returns: { }

  71. 40

    33

    16

    Returns: { }

  72. 31

    14

    23

    Returns: { }

  73. 25

    9

    21

    Returns: { }

  74. 41

    6

    4

    Returns: { }

  75. 4

    1

    1

    Returns: { }

  76. 6

    3

    3

    Returns: {1, 3, 2, 6, 5, 4 }

  77. 50

    20

    5

    Returns: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 14, 13, 20, 19, 18, 17, 16, 25, 24, 23, 22, 21, 30, 29, 28, 27, 26, 35, 34, 33, 32, 31, 40, 39, 38, 37, 36, 45, 44, 43, 42, 41, 50, 49, 48, 47, 46 }

  78. 3

    2

    3

    Returns: { }

  79. 10

    3

    4

    Returns: {2, 1, 6, 5, 4, 3, 10, 9, 8, 7 }

  80. 6

    4

    4

    Returns: { }

  81. 2

    2

    2

    Returns: { }

  82. 50

    13

    13

    Returns: {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38 }

  83. 10

    10

    10

    Returns: { }

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