Grand Dyck consecutive patterns avoiding binary words

Transcription

1 Grand Dyck consecutive patterns avoiding binary words A.Bernini S.Bilotta E.Pergola R.Pinzani UNIVERSITA DEGLI STUDI DI FIRENZE Dipartimento di Matematica e Informatica Permutation Patterns 203 A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 / 37

2 Table of contents Outline Definitions and notations 2 A general approach 3 Main Result 4 Enumeration 5 Generalization 6 Further developments A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 2 / 37

3 Definitions and notations Binary words avoiding a pattern p Let F {0, } be the class of binary words ω such that ω 0 ω for any ω F, ω 0 and ω are the number of zeroes and ones in ω, respectively. We are interested in studying the subclass F [p] F of binary words excluding a given pattern p = p 0... p h {0, } h, i.e. the words ω F [p] that do not admit a sequence of consecutive indices i, i +,..., i + h such that ω i ω i+... ω i+h = p 0 p... p h. R. Sedgewick & P. Flajolet (996): An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading, MA. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 3 / 37

4 Example Definitions and notations p = 0 0 ω = A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 4 / 37

5 Grand Dyck words Definitions and notations Let us denote by G j the set of 2j-length Grand Dyck words. It is well known that a Grand Dyck word δ is a binary string such that δ 0 = δ. In this work, we consider the so called bifix-free Grand Dyck words, i.e. a Grand Dyck word δ is said to be bifix-free if and only if no prefix of δ is a suffix of δ. Let us denote by Gj b the set of 2j-length bifix-free Grand Dyck words and we study the class F [p] with p Gj b, for any fixed j. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

6 A general approach A general approach Let F {0, } be the class of binary words ω such that ω 0 ω for any ω F. For example, in order to generate the language F [p] an ad hoc grammar (depending on the forbidden pattern p) should be defined. Consequently, for each pattern p a different generating function enumerating the words in F [p] must to be computed. Our aim is to determine a constructive algorithm proposing a more unified approach. The algorithm bases on a succession rule which allows us to obtain a generating function for the enumeration of each classes F [p], according to the number of ones. Such enumeration does not depend on the shape of the forbidden pattern p. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 6 / 37

7 Succession rules A general approach A succession rule Ω is a system constituted by an axiom (a), with a N, and a set of productions of the form: (k) (e (k))(e 2 (k))... (e k (k)), k N, e i : N N. A production constructs, for any given label (k), its successors (e (k)), (e 2 (k)),..., (e k (k)). Compact notation: { (a) (k) (e (k))(e 2 (k))... (e k (k)) F. R. K. Chung, R. L. Graham, V. E. Hoggatt & M. Kleimann (978): The number of Baxter permutations. Journal of Combinatorial Theory, Series A 24, pp A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 7 / 37

8 Generating trees A general approach 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 (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. A succession rule describes the growth of a class of combinatorial objects if there exists a bijection between the object of size n and the nodes at level n in the generating tree, then a given object can be coded by the sequence of labels met from the root of the generating tree to the object itself. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

9 Jumping succession rules A general approach 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) (k) (k) (e (k))(e 2 (k))... (e k (k)), j (d (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 27, pp A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 9 / 37

10 A general approach Jumping and marked succession rules A jumping and marked 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) (k) (k) (e (k))(e 2 (k))... (e k (k)), j (d (k))(d 2 (k))... (d k (k)). A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 0 / 37

11 A general approach Jumping and marked succession rules Compact notation: (a) (k) (k) (e (k))(e 2 (k))... (e k (k)), j (d (k))(d 2 (k))... (d k (k)). describes also the behavior for (k): { (k) (e (k))(e 2 (k))... (e k (k)), (k) j (d (k))(d 2 (k))... (d k (k)). having (k) = (k). A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 / 37

12 Example (0) (k) (k) A general approach (k + )(k) ()(0)(0) 2 (k + )(k) ()(0)(0) level 0 (0) () (0) (0) 2 (2) () (0) (0) () (0) (0) () (0) (0) () (0) (0) (k) = (k) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 2 / 37

13 Example (0) (k) (k) A general approach (k + )(k) ()(0)(0) 2 (k + )(k) ()(0)(0) level 0 (0) Cardinality () (0) (0) 3 2 (2) () (0) (0) () (0) (0) () (0) (0) () (0) (0) 7 (k) = (k) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 2 / 37

14 Theorem Main Result Theorem The generating tree of the paths in F [p], where p Gj b, according to the number of rise steps, is isomorphic to the tree having its root labelled (0) and recursively defined by the succession rule: (k) (k + )(k)... ()(0 2 )(0 ) k 0 (0) j (0 2 ) (k) j (k)(k )... ()(0 2 )(0 ) k A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 3 / 37

15 Method Main Result Each node at level n and labelled with (k), k 0, represents a path on the discrete plane having n rise steps and ending at the ordinate k. (0) () (0 ) 2 (0) (2) () (0 2 ) (0 ) () (0 2 ) (0 ) () (0 2 ) (0 ) (0 ) 2 A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 4 / 37

16 Method Main Result Each node at level n and labelled with (k), k 0, represents a path on the discrete plane having n rise steps and ending at the ordinate k. ε () (02) (0) (2) () (0 2 ) () (0 (0) 2 ) (0 ) () (02) (0) (02) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 4 / 37

17 Method Main Result Each node at level n and labelled with (k), k 0, represents a path on the discrete plane having n rise steps and ending at the ordinate k. ε () (0 2 ) (0 ) (2) () (02) (0 ) () (02) (0) () (02) (0) (0 2 ) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 4 / 37

18 Method Main Result Each node at level n and labelled with (k), k 0, represents a path on the discrete plane having n rise steps and ending at the ordinate k. ε () (02) (0) (2) () (0 2 ) () (0 (0) 2 ) (0 ) () (02) (0) (02) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 4 / 37

19 The constructive algorithm: First step ( k) ( k + ) ( k) ( ) ( 0 ) ( 0 ) 2 A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

20 The constructive algorithm: First step ( k) ( k + ) ( k) () ( 02) ( 0) k (k) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

21 The constructive algorithm: First step ( k) (k +) (k) () ( 02) ( 0) k k+ ( k) ( k+) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

22 The constructive algorithm: First step ( k) ( k + ) (k) () ( 0 ) ( ) 2 0 k k+ k ( k) (k+) (k) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

23 The constructive algorithm: First step ( k) ( k + ) ( k)... () ( 02) ( 0 ) k k+ k... ( k) (k+) (k) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

24 The constructive algorithm: First step ( k) ( k + ) ( k) () ( 0 ) ( ) 2 0 k k+ k ( k) (k+) (k) () A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

25 The constructive algorithm: First step ( k) ( k + ) ( k) () (0 2 ) (0 ) k k+ k ( k) (k+) (k) () (0 2) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

26 The constructive algorithm: First step ( k) ( k + ) ( k) ( ) ( ) 0 2 (0 ) k k+ k? (k ) ( k +) ( k) () (02) (0 ) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

27 The constructive algorithm: First step ( k) ( k + ) ( k) () ( 0 ) ( ) 2 0 k k+ k? ( k) (k+) (k) () (0 2) (0) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 5 / 37

28 The constructive algorithm: a marked forbidden pattern We define a marked forbidden pattern p as a pattern p Gj b whose steps cannot be split, that is, they must always be contained all together in that defined sequence. We say that a point is strictly contained in a given marked forbidden pattern p if it is in p and it is different from both its initial point and its last point. We denote a marked forbidden pattern p by drawing its minimal bounding rectangle B. B A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 6 / 37

29 . Main Result The constructive algorithm: First step (0) j (0 ) 2 A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 7 / 37

30 The constructive algorithm: First step (0) j (0 ) 2 j (0) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 7 / 37

31 The constructive algorithm: First step (0) j (0 2 ) j (0) (0 2 ) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 7 / 37

32 The constructive algorithm: First step (0) j (02) j (0) (0 2 ) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 7 / 37

33 The constructive algorithm: First step ( ) j k ( k) ( k-) () (02) (0 ) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

34 The constructive algorithm: First step ( k) j ( k) ( k-) () (02) (0 ) k j ( k) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

35 The constructive algorithm: First step ( k) j ( k) ( k-) () (02) (0 ) k j k (k) ( k) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

36 The constructive algorithm: First step ( k) j ( k) ( k-) () (02) (0 ) k j k k- (k) (k) ( k-) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

37 The constructive algorithm: First step j ( k ) ( k) ( k-)... () (0 ) 2 (0 ) k j k k-... (k) (k) (k-) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

38 The constructive algorithm: First step j ( k ) ( k) ( k-) () (0 ) 2 (0 ) k j k k- (k) (k) (k-) () A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

39 The constructive algorithm: First step j ( k ) ( k) ( k-) () (0 2 ) (0 ) k j k k- (k) (k) (k-) () (02) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

40 The constructive algorithm: First step ( k) j ( k) ( k-) () (02) (0 ) k j k k-? (k) (k) (k-) () (0 2) (0) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

41 The constructive algorithm: First step ( k) j ( k) ( k-) () (02) (0 ) k j k k-? (k) ( k) (k-) () (0 2) (0) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 8 / 37

42 Problem Main Result Lattice paths which end on the x-axis by a rise step are never obtained. j = 2 p = A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 9 / 37

43 Main Result The constructive algorithm: Second step First action in order to obtain the label (0 ) ω = ω = k () (k) j = 2 ω = () ω = vϕ, being ϕ the rightmost suffix in ω beginning from the x-axis with a rise step where each marked forbidden pattern (if ϕ contains at least one of them) has its initial point with positive ordinate. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

44 The constructive algorithm: Second step First action in order to obtain the label (0 ) ϕ ω = ω = (k) k j = 2 ω = () ϕ () ω = vϕ, being ϕ the rightmost suffix in ω beginning from the x-axis with a rise step where each marked forbidden pattern (if ϕ contains at least one of them) has its initial point with positive ordinate. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

45 The constructive algorithm: Second step - case ϕ does not contain any marked forbidden pattern vϕ ϕ A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 2 / 37

46 The constructive algorithm: Second step - case ϕ does not contain any marked forbidden pattern vϕ ϕ A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 2 / 37

47 The constructive algorithm: Second step - case ϕ does not contain any marked forbidden pattern vϕ ϕ vϕ c x x (0 ) ϕ c A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 2 / 37

48 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern Find z the leftmost point in ϕ having highest ordinate and not strictly contained in a marked forbidden pattern. Apply the cut and paste operation. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

49 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern Find z the leftmost point in ϕ having highest ordinate and not strictly contained in a marked forbidden pattern. Apply the cut and paste operation. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

50 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

51 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

52 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

53 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern z z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

54 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern z z z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

55 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern z z z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

56 The constructive algorithm: Second step - case 2 ϕ contains at least one marked forbidden pattern z z z z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

57 The constructive algorithm: Second step - case 2 Cut and paste z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

58 The constructive algorithm: Second step - case 2 Cut and paste z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

59 The constructive algorithm: Second step - case 2 Cut and paste z A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

60 The constructive algorithm: Second step - case 2 Cut and paste (0 ) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

61 The constructive algorithm: Second step - case 2 Reverse of cut and paste (0 ) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

62 The constructive algorithm: Second step - case 2 Reverse of cut and paste m A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

63 The constructive algorithm: Second step - case 2 Reverse of cut and paste m A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

64 The constructive algorithm: Second step - case 2 Reverse of cut and paste m A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

65 The constructive algorithm: Second step - case 2 Reverse of cut and paste m A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

66 The constructive algorithm: Second step - case 2 Reverse of cut and paste A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

67 Example of the complete algorithm (3) (2) () (02 ) (0 ) (2) () (2) (0 2 ) (0 ) A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

68 Proof: idea Main Result 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] having C forbidden patterns, with n rise steps, and (k) as last label of the associated code is also generated having the same shape as ψ but such that the last forbidden pattern is marked if it is not in ψ and vice-versa. ()(0)()(2)() ()(0)()() (0)()(2)() (0)()() A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

69 Enumeration Enumeration Let Z be the set of paths whose instances are coded by a sequence of labels in the generating tree ending by a non-marked zero, S be the set of paths whose instances are coded by a sequence of labels ending by a marked zero, N be the set of paths whose instances are coded by a sequence of labels ending by a non-marked k and M be the set of paths whose instances are coded by a sequence of labels ending by a marked k. Then F [p] = (Z\S) (N\M). Z(x, ) = ω Z xn(ω) y 0 = + 2xZ(x, ) + 2xN(x, ) + x j S(x, ) + 2x j M(x, ), S(x, ) = ω S xn(ω) y 0 = 2xS(x, ) + 2xM(x, ) + x j Z(x, ) + 2x j N(x, ), N(x, y) = ω N xn(ω) y h(ω) = xyz(x, ) + h(ω)+ ω N i= x n(ω)+ y i + h(ω) ω M i= xn(ω)+j y i, M(x, y) = ω M xn(ω) y h(ω) = xys(x, ) + h(ω)+ ω M i= x n(ω)+ y i + h(ω) ω N i= xn(ω)+j y i. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

70 Enumeration Enumeration We obtain the generating function F j (x), j >, for the words ω F [p] according to the number of ones: where F j (x) = y 0 (x) x j ( x j )( + x j 2xy 0 (x)), y 0 (x) = x j + (x j + ) 2 4x. 2x A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

71 Enumeration Enumeration Let us remark that the generating function F j (x) depends only on the number of ones in the forbidden pattern. So, bifix-free Grand Dyck words represent an invariant class for the enumeration of the words which avoid anyone of such a shape path. This means that all the sets in F with n ones avoiding a forbidden pattern p G b j have the same cardinality, independently on the shape of p. Example For any pattern p in G4 b, the first numbers of the sequence enumerating the binary words in F [p], according to the number of ones, are:, 3, 0, 35, 25, 454, 67, 62, 2326, being the associated generating function. F 4 (x) = x 4 x 8 + 2x 4 + 4x 2x( x 3 ) x 8 + 2x 4 + 4x A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

72 Enumeration Why bifix-free forbidden patterns? The method works only for bifix-free forbidden patterns: if p is not bifix-free, then the jumping and marked succession rule should generate paths containing forbidden patterns which could not be eliminated in the right way. For example, when p = ε A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 3 / 37

73 Enumeration Why bifix-free forbidden patterns? The method works only for bifix-free forbidden patterns: if p is not bifix-free, then the jumping and marked succession rule should generate paths containing forbidden patterns which could not be eliminated in the right way. For example, when p = ε......? A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP203 3 / 37

74 Cross-bifix-free patterns Generalization In order to generalize our method we have to consider cross-bifix-free Grand Dyck words. Two distinct bifix-free Grand Dyck words δ, δ are said to be cross-bifix-free if and only if no prefix of δ is also a suffix of δ and vice-versa. A cross-bifix-free set is a set of words where any two words are cross-bifix-free. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

75 Generalization Generalization Given a set P = {p, p 2,, p m } of cross-bifix-free Grand Dyck words, none included in any other, such that each p l having j l as the number of its s, l m, then the growth of the set F [P], according to the number of ones, can be described by the following jumping and marked succession rule: (0) (k) (k + )(k) ()(0 2 )(0 ) k 0 (0) j (0 2 ) (k) j (k)(k ) ()(0 2 )(0 ) k (0) j2 (0 2 ) (k) j2 (k)(k ) ()(0 2 )(0 ) k. (0) jm (0 2 ) (k) jm (k)(k ) ()(0 2 )(0 ) k A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

76 Generalization Generalization By using the previous notation, we have: Z(x, ) = + 2xZ(x, ) + 2xN(x, ) + (x j + + x jm )S(x, ) + 2(x j + + x jm )M(x, ), S(x, ) = 2xS(x, ) + 2xM(x, ) + (x j + + x jm )Z(x, ) + 2(x j + + x jm )N(x, ), N(x, y) = xyz(x, )+ ω N h(ω)+ i= x n(ω)+ y i + h(ω) ω M i= xn(ω)+j y i + + h(ω) ω M i= xn(ω)+jm y i, M(x, y) = xys(x, )+ ω M h(ω)+ i= x n(ω)+ y i + h(ω) ω N i= xn(ω)+j y i + + h(ω) ω N i= xn(ω)+jm y i. A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

77 Generalization Generalization We obtain the generating function F j,...,j m (x) for the words ω F [P] according to the number of ones: F j,,j m (x) = where y 0 (x) x j x jm ( x j x jm )( + x j + + x jm 2xy 0 (x)), y 0 (x) = + x j + + x jm (x j + + x jm + ) 2 4x. 2x A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

78 Further developments Further developments. Study other forbidden patterns p 2. Expand the alphabet {0, } 3. Find a unified approach for pattern avoidance construction and enumeration A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

79 Further developments Further developments. Study other forbidden patterns p 2. Expand the alphabet {0, } 3. Find a unified approach for pattern avoidance construction and enumeration A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

80 Further developments Further developments. Study other forbidden patterns p 2. Expand the alphabet {0, } 3. Find a unified approach for pattern avoidance construction and enumeration A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37

81 Thanks for your attention A.Bernini S.Bilotta E.Pergola R.Pinzani (DiMaI) Grand Dyck patterns avoiding binary words PP / 37