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Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani Dipartimento di Sistemi e Informatica Universit` a degli Studi di Firenze WORDS 2011 Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS

  1. Definitions and notations Succession rules Generating trees The rule Ω can be represented by means of a generating tree , that is a rooted tree whose vertices are the labels of Ω; where ( a ) is the label of the root and each node labelled ( k ) has k sons labelled ( e 1 ( k )) , ( e 2 ( k )) , . . . , ( e k ( k )), respectively. As usual, the root lies at level 0, and a node lies at level n if its parent lies at level n − 1. If a succession rule describes the growth of a class of combinatorial objects, then a given object can be coded by the sequence of labels met from the root of the generating tree to the object itself. Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 8 / 27

  2. Definitions and notations Succession rules Jumping succession rules A jumping succession rule is a set of productions acting on the objects of a class and producing sons at different levels. Compact notation:  ( a )   1 ( k ) � ( e 1 ( k ))( e 2 ( k )) . . . ( e k ( k )) , j  ( k ) � ( d 1 ( k ))( d 2 ( k )) . . . ( d k ( k )) .  • L. Ferrari, E. Pergola, R. Pinzani & S. Rinaldi (2003): Jumping succession rules and their generating functions . Discrete Mathematics 271, pp. 29-50. Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 9 / 27

  3. Definitions and notations Succession rules Jumping succession rules A jumping succession rule is a set of productions acting on the objects of a class and producing sons at different levels. Compact notation:  ( a )   1 ( k ) � ( e 1 ( k ))( e 2 ( k )) . . . ( e k ( k )) , j  ( k ) � ( d 1 ( k ))( d 2 ( k )) . . . ( d k ( k )) .  • L. Ferrari, E. Pergola, R. Pinzani & S. Rinaldi (2003): Jumping succession rules and their generating functions . Discrete Mathematics 271, pp. 29-50. Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 9 / 27

  4. Definitions and notations Succession rules Jumping succession rules A jumping succession rule is a set of productions acting on the objects of a class and producing sons at different levels. Compact notation:  ( a )   1 ( k ) � ( e 1 ( k ))( e 2 ( k )) . . . ( e k ( k )) , j  ( k ) � ( d 1 ( k ))( d 2 ( k )) . . . ( d k ( k )) .  • L. Ferrari, E. Pergola, R. Pinzani & S. Rinaldi (2003): Jumping succession rules and their generating functions . Discrete Mathematics 271, pp. 29-50. Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 9 / 27

  5. Definitions and notations Succession rules Marked jumping succession rules A marked jumping succession rule is a jumping succession rule where marked labels are considered together with usual ones. In this way a generating tree can support negative values if we consider a node labelled ( k ) as opposed to a node labelled ( k ) lying on the same level. Compact notation:  ( a )   1 ( k ) � ( e 1 ( k ))( e 2 ( k )) . . . ( e k ( k )) , j  ( k ) � ( d 1 ( k ))( d 2 ( k )) . . . ( d k ( k )) .  Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 10 / 27

  6. Definitions and notations Succession rules Marked jumping succession rules A marked jumping succession rule is a jumping succession rule where marked labels are considered together with usual ones. In this way a generating tree can support negative values if we consider a node labelled ( k ) as opposed to a node labelled ( k ) lying on the same level. Compact notation:  ( a )   1 ( k ) � ( e 1 ( k ))( e 2 ( k )) . . . ( e k ( k )) , j  ( k ) � ( d 1 ( k ))( d 2 ( k )) . . . ( d k ( k )) .  Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 10 / 27

  7. Definitions and notations Succession rules Example  (0)   1 � (0) 2 (1) · · · ( k + 1) ( k ) 2  � (0) 2 (1) · · · ( k + 1) ( k )  level (0) 0 1 (0) (0) (1) (0) (0) (1) (0) (0) (1) (0) (0) (1) (2) (0) (0) (1) 2 ( k ) = ( k ) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 11 / 27

  8. Definitions and notations Succession rules Example  (0)   1 � (0) 2 (1) · · · ( k + 1) ( k ) 2  � (0) 2 (1) · · · ( k + 1) ( k )  level Cardinality (0) 0 1 1 (0) (0) (1) 3 (0) (0) (1) (0) (0) (1) (0) (0) (1) (2) (0) (0) (1) 2 7 ( k ) = ( k ) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 11 / 27

  9. p ( j ) = 1 j +10 j , j ≥ 1 Results Results Theorem The generating tree of the class F [ p ( j )] , for any fixed p ( j ) = 1 j +1 0 j , j ≥ 1, according to the number of ones, is isomorphic to the tree having its root labelled (0) and recursively defined by the following succession rule: � 1 � (0) 2 (1) . . . ( k + 1) , ( k ) j +1 � (0) 2 (1) . . . ( k + 1) . ( k ) • S. Bilotta, D. Merlini, E. Pergola & R. Pinzani (2010): Binary words avoiding a pattern and marked succession rule. In: Lattice Path Combinatorics and Applications, Siena. Available on line arXiv:1103.5689. Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 12 / 27

  10. p ( j ) = 1 j +10 j , j ≥ 1 Results Results Theorem The generating tree of the class F [ p ( j )] , for any fixed p ( j ) = 1 j +1 0 j , j ≥ 1, according to the number of ones, is isomorphic to the tree having its root labelled (0) and recursively defined by the following succession rule: � 1 � (0) 2 (1) . . . ( k + 1) , ( k ) j +1 � (0) 2 (1) . . . ( k + 1) . ( k ) • S. Bilotta, D. Merlini, E. Pergola & R. Pinzani (2010): Binary words avoiding a pattern and marked succession rule. In: Lattice Path Combinatorics and Applications, Siena. Available on line arXiv:1103.5689. Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 12 / 27

  11. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) ( k ) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  12. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) ( k ) ( k ) k ω = ( k ) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  13. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) ( k ) ( k + 1) k ω = ( k ) 1 k + 1 ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  14. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) ( k ) ( k ) k ω = ( k ) 1 , k + 1 k ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  15. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) . ( k ) k ω = ( k ) 1 . , , k + 1 k ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  16. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) . ( k ) k ω = ( k ) 1 . . , , k + 1 k ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  17. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) . ( k ) k ω = ( k ) 1 . . . , , k + 1 k ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  18. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) ( k ) (1) k ω = ( k ) 1 . . . , , , k + 1 k 1 (1) ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  19. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) ( k ) (0) k ω = ( k ) 1 . . . , , , , k + 1 k 1 (0) (1) ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  20. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) ( k ) (0) k ω = ( k ) 1 ? . . . , , , , , k + 1 k 1 (0) (1) ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  21. Constructing F [ p ] The algorithm The constructive algorithm: first step 1 (0)(0)(1) . . . ( k )( k + 1) ( k ) (0) k ω = ( k ) 1 ? . . . , , , , , k + 1 k 1 (0) (1) ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 13 / 27

  22. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) j + 1 ( k ) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  23. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) j + 1 ( k ) ( k ) k ω = ( k ) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  24. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) 2 ( k ) ( k + 1) j = 1 k ω = ( k ) 2 k + 1 ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  25. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) 2 ( k ) ( k ) j = 1 k ω = ( k ) 2 k + 1 k , ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  26. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) . 2 ( k ) j = 1 k ω = ( k ) 2 . k + 1 , k , ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  27. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) . 2 ( k ) j = 1 k ω = ( k ) 2 . . k + 1 , k , ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  28. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) . 2 ( k ) j = 1 k ω = ( k ) 2 . . . k + 1 , k , ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  29. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) 2 ( k ) (1) j = 1 k ω = ( k ) 2 . . . k + 1 , , k , 1 (1) ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  30. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) 2 ( k ) (0) (1) j = 1 k ω = ( k ) 2 . . . k + 1 , , , k , 1 (0) (1) ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  31. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) 2 ( k ) (0) (1) j = 1 k ω = ( k ) 2 . . . ? k + 1 , , , , k , 1 (0) (1) ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  32. Constructing F [ p ] The algorithm The constructive algorithm: second step (0)(0)(1) . . . ( k )( k + 1) 2 ( k ) (0) (1) j = 1 k ω = ( k ) 2 . . . ? k + 1 , , , , k , 1 (0) (1) ( k ) ( k + 1) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 14 / 27

  33. Constructing F [ p ] The algorithm Problem Lattice paths which either do not contain marked forbidden pattern in its rightmost suffix and end on the x -axis by a rise step or have the rightmost marked point with ordinate less than or equal to j , are never obtained. j = 2 p = Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 15 / 27

  34. Constructing F [ p ] The algorithm The constructive algorithm: third step First action in order to obtain the label (0) 1 ω ′ = 1 (1) k ω = j + 1 = 2 ( k ) ω ′ = 1 (1) ω ′ = v ϕ , being ϕ is the rightmost suffix in ω ′ beginning from the x -axis and strictly remaining above the x -axis. Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 16 / 27

  35. Constructing F [ p ] The algorithm The constructive algorithm: third step First action in order to obtain the label (0) ϕ 1 ω ′ = 1 (1) k ω = ϕ j + 1 = 2 ( k ) ω ′ = 1 (1) ω ′ = v ϕ , being ϕ is the rightmost suffix in ω ′ beginning from the x -axis and strictly remaining above the x -axis. Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 16 / 27

  36. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 1 ϕ does not contain any marked point v ϕ ϕ 1 Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 17 / 27

  37. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 1 ϕ does not contain any marked point v ϕ ϕ 1 Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 17 / 27

  38. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 1 ϕ does not contain any marked point v ϕ v ϕ c x ϕ 1 x ϕ c (0) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 17 / 27

  39. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  40. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  41. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r t Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  42. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r t s Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  43. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z t s Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  44. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z t s Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  45. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  46. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s r Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  47. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s t r Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  48. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s t s r Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  49. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t s r Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  50. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t s r Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  51. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t z s r Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  52. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t z s r r Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  53. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t z s r r t Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  54. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t z s r r t s Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  55. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t z s r r z ≡ t s Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  56. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t z s r r z ≡ t s Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  57. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 ϕ contains at least one marked point r z z t s z ≡ t z s r r z z ≡ t s Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 18 / 27

  58. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Cut and paste z Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 19 / 27

  59. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Cut and paste z Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 19 / 27

  60. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Cut and paste z Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 19 / 27

  61. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Cut and paste z Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 19 / 27

  62. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Cut and paste z Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 19 / 27

  63. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Cut and paste z Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 19 / 27

  64. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Cut and paste (0) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 19 / 27

  65. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste (0) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  66. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste m Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  67. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste m Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  68. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste m Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  69. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste m Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  70. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste m Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  71. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste m Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  72. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste m Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  73. Constructing F [ p ] The algorithm The constructive algorithm: third step - case 2 Reverse of cut and paste Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 20 / 27

  74. Constructing F [ p ] The algorithm Example of the complete algorithm (3) (2) (1) (0) (0) (2) (3) (2) (1) (0) (0) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 21 / 27

  75. Constructing F [ p ] The algorithm The generating tree (2) (3) (2) (1) (0) (0) (1) (2) (1) (0) (0) (0) (1) (0) (0) (1) (1) (0) (0) (0) (2) (1) (0) (0) (1) (2) (1) (0) (0) (0) (1) (0) (0) (0) (0) (1) (0) (0) (1) (0) (0) (1) (2) (1) (0) (0) (0) (1) (0) (0) ε (0) (0) (1) (0) (0) (1) (0) (0) (1) (2) (1) (0) (0) (0) (1) (0) (0) (0) (1) (0) (0) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 22 / 27

  76. Constructing F [ p ] Proof: idea Proof: idea We have to show that the described algorithm is a construction for the set F [ p ] according to the number of rise steps. This means that all the paths in F with n rise steps are obtained. Moreover, for each obtained path ψ in F \ F [ p ] , having C forbidden patterns, with n rise steps and ( k ) as last label of the associated code, a path ψ ′ in F \ F [ p ] with n rise steps, C forbidden patterns and ( k ) as last label of the associated code is also generated having the same form as ψ but such that the last forbidden pattern is marked if it is not in ψ and vice-versa. (1) (0) (0) (0) (1) (0) (0) (1) (0) (0) (1) (0) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 23 / 27

  77. p ( j ) = 1 j +10 j − → p ( j , i ) = 1 j 0 i , 0 < i < j Final results In this work, we study the class F [ p ] focusing on the generalization of the fixed forbidden pattern p . p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j , i ) = 1 j 0 i , 0 < i < j Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 24 / 27

  78. p ( j ) = 1 j +10 j − → p ( j , i ) = 1 j 0 i , 0 < i < j Final results In this work, we study the class F [ p ] focusing on the generalization of the fixed forbidden pattern p . p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j , i ) = 1 j 0 i , 0 < i < j  (0)   1 � (0) 2 (1) · · · ( k )( k + 1) ( k ) j +1  � (0) 2 (1) · · · ( k )( k + 1) ( k )  Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 24 / 27

  79. p ( j ) = 1 j +10 j − → p ( j , i ) = 1 j 0 i , 0 < i < j Final results In this work, we study the class F [ p ] focusing on the generalization of the fixed forbidden pattern p . p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j , i ) = 1 j 0 i , 0 < i < j  (0)   1  � (0) 2 (1) · · · ( k )( k + 1) ( k )    j +1  � (0) 2 (1) · · · ( k )( k + 1) ( k )  Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 24 / 27

  80. p ( j ) = 1 j +10 j − → p ( j , i ) = 1 j 0 i , 0 < i < j Final results In this work, we study the class F [ p ] focusing on the generalization of the fixed forbidden pattern p . p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j , i ) = 1 j 0 i , 0 < i < j  (0)   1 � (0) 2 (1) · · · ( k )( k + 1)  ( k )         ( k + 1)         ( k )      .   .  j +1    ( k )  �  .       (1)         (0)        (0)  Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 24 / 27

  81. p ( j ) = 1 j +10 j − → p ( j , i ) = 1 j 0 i , 0 < i < j Final results In this work, we study the class F [ p ] focusing on the generalization of the fixed forbidden pattern p . p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j , i ) = 1 j 0 i , 0 < i < j  (0)   1 � (0) 2 (1) · · · ( k )( k + 1)  ( k )         ( k + 1)         ( k )      .   .  j +1    ( k )  �  .       (1)         (0)        (0)  Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 24 / 27

  82. p ( j ) = 1 j +10 j − → p ( j , i ) = 1 j 0 i , 0 < i < j Final results In this work, we study the class F [ p ] focusing on the generalization of the fixed forbidden pattern p . p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j , i ) = 1 j 0 i , 0 < i < j  (0)   1 � (0) 2 (1) · · · ( k )( k + 1)  ( k )          ( k + 1)        ( k )   .   j +1    ( k ) .  �      .       (1) − → (0)         (0) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 24 / 27

  83. p ( j ) = 1 j +10 j − → p ( j , i ) = 1 j 0 i , 0 < i < j Final results In this work, we study the class F [ p ] focusing on the generalization of the fixed forbidden pattern p . p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j , i ) = 1 j 0 i , 0 < i < j  (0)   1 � (0) 2 (1) · · · ( k )( k + 1)  ( k )          ( k + 1)        ( k )   .   j +1    ( k ) .  �      .       (1) − → (0)         (0) Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 24 / 27

  84. p ( j ) = 1 j +10 j − → p ( j , i ) = 1 j 0 i , 0 < i < j Final results In this work, we study the class F [ p ] focusing on the generalization of the fixed forbidden pattern p . p ( j ) = 1 j +1 0 j , j ≥ 1 p ( j , i ) = 1 j 0 i , 0 < i < j p ( j , i ) = 1 j 0 i , 0 < i < j  (0)   1  � (0) 2 (1) · · · ( k )( k + 1) ( k )     j � (0) j − i +1 − a (1) j − i − a (2) j − i − 1 − a . . . ( k )       . . . ( j − i − 1 − a ) 2 ( j − i − a ) . . . ( k + j − i )  Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS 2011 24 / 27

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