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Enumerating permutations sortable by k passes through a pop-stack Anders Claesson Bjarki gst Gumundsson University of Iceland 1 / 20 Theorem (Knuth, 1968) 3124 A permutation is sortable by a stack if and only if avoids 231 .

  1. Enumerating permutations sortable by k passes through a pop-stack Anders Claesson Bjarki Ágúst Guðmundsson University of Iceland 1 / 20

  2. Theorem (Knuth, 1968) 3124 A permutation π is sortable by a stack if and only if π avoids 231 . Theorem (Avis & Newborn, 1981) 3124 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . 2 / 20

  3. Theorem (Knuth, 1968) 124 A permutation π is sortable by a stack if and only if π avoids 231 . 3 Theorem (Avis & Newborn, 1981) 124 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . 3 2 / 20

  4. Theorem (Knuth, 1968) 24 A permutation π is sortable by a stack 1 if and only if π avoids 231 . 3 Theorem (Avis & Newborn, 1981) 24 A permutation π is sortable by a pop-stack 1 if and only if π avoids 231 and 321 . 3 2 / 20

  5. Theorem (Knuth, 1968) 1 24 A permutation π is sortable by a stack if and only if π avoids 231 . 3 Theorem (Avis & Newborn, 1981) 13 24 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . 2 / 20

  6. Theorem (Knuth, 1968) 1 4 A permutation π is sortable by a stack 2 if and only if π avoids 231 . 3 Theorem (Avis & Newborn, 1981) 13 4 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . 2 2 / 20

  7. Theorem (Knuth, 1968) 12 4 A permutation π is sortable by a stack if and only if π avoids 231 . 3 Theorem (Avis & Newborn, 1981) 132 4 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . 2 / 20

  8. Theorem (Knuth, 1968) 123 4 A permutation π is sortable by a stack if and only if π avoids 231 . Theorem (Avis & Newborn, 1981) 132 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . 4 2 / 20

  9. Theorem (Knuth, 1968) 123 A permutation π is sortable by a stack if and only if π avoids 231 . 4 Theorem (Avis & Newborn, 1981) 1324 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . 2 / 20

  10. Theorem (Knuth, 1968) 1234 A permutation π is sortable by a stack if and only if π avoids 231 . Theorem (Avis & Newborn, 1981) 1324 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . 2 / 20

  11. Theorem (Knuth, 1968) 1234 A permutation π is sortable by a stack if and only if π avoids 231 . Theorem (Avis & Newborn, 1981) 1324 A permutation π is sortable by a pop-stack if and only if π avoids 231 and 321 . Theorem (Pudwell & Smith, 2017) A permutation π is sortable by two passes through a pop-stack if and only if π avoids 2341 , 3412 , 3421 , 4123 , 4231 , 4312 , 4¯ 1352 and 413¯ 52 . 2 / 20

  12. Theorem (Knuth, 1968) √ 1 − 1 − 4 x A permutation π is sortable by a stack if and only if π avoids 231 . 2 x Theorem (Avis & Newborn, 1981) A permutation π is sortable by a pop-stack x − 1 if and only if π avoids 231 and 321 . 2 x − 1 Theorem (Pudwell & Smith, 2017) A permutation π is sortable by two passes x 3 + x 2 + x − 1 through a pop-stack if and only if π avoids 2 x 3 + x 2 + 2 x − 1 2341 , 3412 , 3421 , 4123 , 4231 , 4312 , 4¯ 1352 and 413¯ 52 . 2 / 20

  13. Stacks vs pop-stacks A stack is a LIFO data structure with two operations: ◮ Push: Add an element to the top of the stack. ◮ Pop: Remove the top-most element from the stack. A pop-stack is a LIFO data structure with two operations: ◮ Push: Add an element to the top of the stack. ◮ Pop: Remove all elements from the stack. We’ll insist that elements on the stack are increasing when read from top to bottom and sort greedily w.r.t. the push operation. 3 / 20

  14. How many permutations are sortable by k passes through a pop-stack? x − 1 k = 1 2 x − 1 x 3 + x 2 + x − 1 k = 2 2 x 3 + x 2 + 2 x − 1 k ≥ 3 Rational? 4 / 20

  15. 5 1 2 4 7 8 6 3 9 5 / 20

  16. 5 1 2 4 7 8 6 3 9 5 / 20

  17. 5 1 2 4 7 8 6 3 9 5 / 20

  18. 5 1 2 4 7 8 6 3 9 1 5 5 / 20

  19. 5 1 2 4 7 8 6 3 9 1 5 5 / 20

  20. 5 1 2 4 7 8 6 3 9 1 5 2 5 / 20

  21. 5 1 2 4 7 8 6 3 9 1 5 2 5 / 20

  22. 5 1 2 4 7 8 6 3 9 1 5 2 4 5 / 20

  23. 5 1 2 4 7 8 6 3 9 1 5 2 4 5 / 20

  24. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 5 / 20

  25. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 5 / 20

  26. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 5 / 20

  27. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 5 / 20

  28. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 5 / 20

  29. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 5 / 20

  30. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 5 / 20

  31. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 5 / 20

  32. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 5 / 20

  33. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 5 / 20

  34. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 5 / 20

  35. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 5 / 20

  36. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 5 / 20

  37. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 5 / 20

  38. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 5 / 20

  39. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 5 / 20

  40. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 5 / 20

  41. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 5 / 20

  42. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 5 / 20

  43. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 5 / 20

  44. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 5 / 20

  45. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 5 / 20

  46. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 5 / 20

  47. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 5 / 20

  48. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 5 / 20

  49. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 5 / 20

  50. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 5 / 20

  51. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 5 / 20

  52. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 5 / 20

  53. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 5 / 20

  54. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 5 / 20

  55. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 5 / 20

  56. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 5 / 20

  57. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 6 7 5 / 20

  58. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 6 7 5 / 20

  59. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 6 7 8 5 / 20

  60. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 6 7 8 5 / 20

  61. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 6 7 8 9 5 / 20

  62. 5 1 2 4 7 8 6 3 9 1 5 2 4 7 3 6 8 9 1 2 5 4 3 7 6 8 9 1 2 3 4 5 6 7 8 9 ◮ A sorting trace of length 9 and order 3. ◮ Numbers within a block are in decreasing order. ◮ Adjacent numbers in different blocks form an ascent. ◮ Each perm is the “blockwise reversal” of the one above. ◮ The last permutation is the identity. 5 / 20

  63. 7 5 2 4 9 1 8 6 3 2 5 7 4 1 9 3 6 8 2 5 1 4 7 3 9 6 8 2 1 5 4 3 7 6 9 8 1 2 3 4 5 6 7 8 9 A sorting trace Its sorting plan a d d a a d a d d a a a a d d a d a a a 0 , 9 , 10 , 5 , 5 , 10 , 5 , 10 , 9 , 0 a a d a a d a d a a a d a d d a d a d a The same sorting plan . . . and its encoding 6 / 20

  64. 7 5 2 4 9 1 8 6 3 2 5 7 4 1 9 3 6 8 2 5 1 4 7 3 9 6 8 2 1 5 4 3 7 6 9 8 1 2 3 4 5 6 7 8 9 A sorting trace Its sorting plan 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 , 9 , 10 , 5 , 5 , 10 , 5 , 10 , 9 , 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 0 The same sorting plan . . . and its encoding 6 / 20

  65. Basic bijections k -pop-stack-sortable permutations sort using pop-stack take topmost permutation sorting traces drop numbers append id, then blockwise rev. sorting plans Bijection between k -pop-stack-sortable permutations of [ n ] and sorting plans of length n and order k 7 / 20

  66. An operation array of length 9 and order 3: 8 / 20

  67. An operation array of length 9 and order 3: ◮ Assume there is a sorting trace with this operation array. 8 / 20

  68. An operation array of length 9 and order 3: 1 2 3 4 5 6 7 8 9 ◮ Assume there is a sorting trace with this operation array. ◮ The last permutation must be the identity. 8 / 20

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