An extension of the Pascal triangle and the Sierpiński gasket to finite words. Joint work with Julien Leroy and Michel Rigo (ULiège)

An extension of the Pascal triangle and the Sierpiński gasket to finite words Joint work with Julien Leroy and Michel Rigo (ULiège) Manon Stipulanti (ULiège) FRIA grantee LaBRI, Bordeaux (France) December 5, 27

Classical Pascal triangle ( m ) k k 2 3 4 5 6 7 2 2 m 3 3 3 4 4 6 4 5 5 5 6 6 5 2 5 6 7 7 2 35 35 2 7.... Usual binomial coefficients of integers: ( ) m m! = k (m k)! k! Pascal s rule: vide ( ) ( ) ( ) m m m = + k k k Generalized Pascal triangles Manon Stipulanti (ULiège) 2

A specific construction Grid: intersection between N 2 and [, 2 n ] [, 2 n ] 2 n 2 n ( ) ( ) ( ) 2 n ( ) ( ) ( ) 2 n....... 2 n ( 2 n ) ( 2 n ) ( 2 ) n 2 n 2 n N 2 [, 2 n ] 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 3

Color the grid: Color the first 2 n rows and columns of the Pascal triangle (( ) ) m mod 2 k in white if ( ) m k mod 2 black if ( ) m k mod 2 m,k<2 n Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Color the grid: Color the first 2 n rows and columns of the Pascal triangle (( ) ) m mod 2 k m,k<2 n in white if ( ) m k mod 2 black if ( ) m k mod 2 Normalize by a homothety of ratio /2 n (bring into [, ] [, ]) sequence belonging to [, ] [, ] Generalized Pascal triangles Manon Stipulanti (ULiège) 4

What happens for n {, } n = fantome Generalized Pascal triangles Manon Stipulanti (ULiège) 5

What happens for n {, } n = fantome ( ) Generalized Pascal triangles Manon Stipulanti (ULiège) 5

What happens for n {, } n = fantome ( ) n = fantome Generalized Pascal triangles Manon Stipulanti (ULiège) 5

What happens for n {, } n = fantome ( ) n = fantome 2 ( ) ( ) ( ) ( ) 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 5

What happens for n {, } n = fantome ( ) n = fantome 2 ( ) ( ) ( ) ( ) 2 2 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 5

What happens for n {, } n = fantome ( ) n = fantome 2 ( ) ( ) ( ) ( ) 2 2 2 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 5 2

The first six elements of the sequence 2 2 2 2 2 2 2 3 2 4 2 5 2 3 2 4 2 5 Generalized Pascal triangles Manon Stipulanti (ULiège) 6

The tenth element of the sequence 2 9 2 9 Generalized Pascal triangles Manon Stipulanti (ULiège) 7

The Sierpiński gasket Generalized Pascal triangles Manon Stipulanti (ULiège) 8

Folklore fact The latter sequence converges to the Sierpiński gasket when n tends to infinity (for the Hausdorff distance). Generalized Pascal triangles Manon Stipulanti (ULiège) 9

Folklore fact The latter sequence converges to the Sierpiński gasket when n tends to infinity (for the Hausdorff distance). Definitions: ɛ-fattening of a subset S R 2 [S] ɛ = x S B(x, ɛ) (H(R 2 ), d h ) complete space of the non-empty compact subsets of R 2 equipped with the Hausdorff distance d h d h (S, S ) = min{ɛ R S [S ] ɛ and S [S] ɛ } Generalized Pascal triangles Manon Stipulanti (ULiège) 9

Remark (von Haeseler, Peitgen, Skordev, 992) The sequence also converges for other modulos. For instance, the sequence converges when the Pascal triangle is considered modulo p s where p is a prime and s is a positive integer. Generalized Pascal triangles Manon Stipulanti (ULiège)

This talk... Part I: Generalized Pascal triangle Part II: Counting Subword Occurrences Part III: Behavior of the Summatory Function Generalized Pascal triangles Manon Stipulanti (ULiège)

Part I: Generalized Pascal triangle

New idea Replace usual binomial coefficients of integers by binomial coefficients of finite words Generalized Pascal triangles Manon Stipulanti (ULiège) 3

Binomial coefficient of finite words Definition: A finite word is a finite sequence of letters belonging to a finite set called alphabet. Example:, {, } Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Binomial coefficient of finite words Definition: A finite word is a finite sequence of letters belonging to a finite set called alphabet. Example:, {, } Binomial coefficient of words (Lothaire, 997) Let u, v be two finite words. The binomial coefficient ( u v) of u and v is the number of times v occurs as a subsequence of u (meaning as a scattered subword). Example: u = v = ( ) = 6 Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Binomial coefficient of finite words Definition: A finite word is a finite sequence of letters belonging to a finite set called alphabet. Example:, {, } Binomial coefficient of words (Lothaire, 997) Let u, v be two finite words. The binomial coefficient ( u v) of u and v is the number of times v occurs as a subsequence of u (meaning as a scattered subword). Example: u = v = occurrence ( ) = 6 Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Binomial coefficient of finite words Definition: A finite word is a finite sequence of letters belonging to a finite set called alphabet. Example:, {, } Binomial coefficient of words (Lothaire, 997) Let u, v be two finite words. The binomial coefficient ( u v) of u and v is the number of times v occurs as a subsequence of u (meaning as a scattered subword). Example: u = v = 2 occurrences ( ) = 6 Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Binomial coefficient of finite words Definition: A finite word is a finite sequence of letters belonging to a finite set called alphabet. Example:, {, } Binomial coefficient of words (Lothaire, 997) Let u, v be two finite words. The binomial coefficient ( u v) of u and v is the number of times v occurs as a subsequence of u (meaning as a scattered subword). Example: u = v = 3 occurrences ( ) = 6 Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Binomial coefficient of finite words Definition: A finite word is a finite sequence of letters belonging to a finite set called alphabet. Example:, {, } Binomial coefficient of words (Lothaire, 997) Let u, v be two finite words. The binomial coefficient ( u v) of u and v is the number of times v occurs as a subsequence of u (meaning as a scattered subword). Example: u = v = 4 occurrences ( ) = 6 Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Binomial coefficient of finite words Definition: A finite word is a finite sequence of letters belonging to a finite set called alphabet. Example:, {, } Binomial coefficient of words (Lothaire, 997) Let u, v be two finite words. The binomial coefficient ( u v) of u and v is the number of times v occurs as a subsequence of u (meaning as a scattered subword). Example: u = v = 5 occurrences ( ) = 6 Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Binomial coefficient of finite words Definition: A finite word is a finite sequence of letters belonging to a finite set called alphabet. Example:, {, } Binomial coefficient of words (Lothaire, 997) Let u, v be two finite words. The binomial coefficient ( u v) of u and v is the number of times v occurs as a subsequence of u (meaning as a scattered subword). Example: u = v = 6 occurrences ( ) = 6 Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Remark: Natural generalization of binomial coefficients of integers With a one-letter alphabet {a} ( ) a m = a k ( m times {}}{) a a = a } {{ a} k times ( ) m k m, k N Generalized Pascal triangles Manon Stipulanti (ULiège) 5

Theoretical framework Definitions: rep 2 (n) greedy base-2 expansion of n N > beginning by rep 2 () := ε where ε is the empty word n rep 2 (n) ε 2 2 2 + 2 3 2 + 2 4 2 2 + 2 + 2 5 2 2 + 2 + 2 6 2 2 + 2 + 2... {ε} {, } Generalized Pascal triangles Manon Stipulanti (ULiège) 6

Generalized Pascal triangle in base 2 ( u v) v ε ε u 2 2 2 2 2 3 3.... Binomial coefficient of finite words: ( u v) Rule (not local): ( ) ( ) ( ) ua u u = + δ a,b vb vb v Generalized Pascal triangles Manon Stipulanti (ULiège) 7

Generalized Pascal triangle in base 2 v ε ε u 2 2 2 2 2 3 3.... The classical Pascal triangle Generalized Pascal triangles Manon Stipulanti (ULiège) 7

Questions: After coloring and normalization can we expect the convergence to an analogue of the Sierpiński gasket? Could we describe this limit object? Generalized Pascal triangles Manon Stipulanti (ULiège) 8

Same construction Grid: intersection between N 2 and [, 2 n ] [, 2 n ] 2 n 2 n ( rep2 () rep 2 () ) ( rep2 ) () rep 2 () ( rep2 ) () rep 2 (2 n ) ( rep2 () rep 2 () ) ( rep2 ) () rep 2 () ( rep2 ) () rep 2 (2 n )....... 2 n ( rep2 (2 n ) rep 2 () ) ( rep2 (2 n ) rep 2 () ) ( rep2 ) (2 n ) rep 2 (2 n ) 2 n N 2 [, 2 n ] 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 9

Color the grid: Color the first 2 n rows and columns of the generalized Pascal triangle (( ) ) rep2 (m) mod 2 rep 2 (k) m,k<2 n in white if ( rep 2 (m) rep 2 (k) ) mod 2 black if ( rep 2 (m) rep 2 (k) ) mod 2 Normalize by a homothety of ratio /2 n (bring into [, ] [, ]) sequence (U n ) n belonging to [, ] [, ] U n := 2 n u,v {ε} {,} s.t. ( u v) mod 2 Q := [, ] [, ] {(val 2 (v), val 2 (u)) + Q} Generalized Pascal triangles Manon Stipulanti (ULiège) 2

The elements U,..., U 5 ε ε ε 2 2 ε 3 4 5 3 4 5 Generalized Pascal triangles Manon Stipulanti (ULiège) 2

The element U 2 /4 2/4 3/4 ε /4 2/4 3/4 ε ε, /4, 2/4 = /2, 3/4 w {ε} {, } with w 2 val 2(w) 2 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 22

Lines of different slopes... Generalized Pascal triangles Manon Stipulanti (ULiège) 23 The element U 9 9 9

The ( ) condition ( ) (u, v) satisfies ( ) iff ( u ( v u v ) u, v ε ) mod ( 2 = = u v) Example: (u, v) = (, ) satisfies ( ) ( ) = ( ) ( = ) = Generalized Pascal triangles Manon Stipulanti (ULiège) 24

Lemma: Completion (u, v) satisfies ( ) (u, v), (u, v) satisfy ( ) Proof: ( ) ( ) u u = v v }{{} = since ( ) ) ( > or u + ( ) u v }{{} mod 2 mod 2 If ( u v v) >, then v is a subsequence of u, which contradicts ( ). Same proof for (u, v). Generalized Pascal triangles Manon Stipulanti (ULiège) 25

Example: u =, v = U 3 U 4 U 5 Creation of segments of slope Endpoint (3/8, 5/8) = (val 2 ()/2 3, val 2 ()/2 3 ) Length 2 2 3 S u,v [, ] [/2, ] endpoint (val 2 (v)/2 u, val 2 (u)/2 u ) length 2 2 u Generalized Pascal triangles Manon Stipulanti (ULiège) 26

Definition: Set of segments of slope A := (u,v) satisfying( ) S u,v [, ] [/2, ] Generalized Pascal triangles Manon Stipulanti (ULiège) 27

Modifying the slope Example: (, ) satisfies ( ) Segment S, endpoint (/2, /2) length 2 2 2 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 28

Modifying the slope Example: (, ) satisfies ( ) Segment S, endpoint (/2, /2) length 2 2 4 2 4 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 28

Modifying the slope Example: (, ) satisfies ( ) Segment S, endpoint (/2, /2) length 2 2 8 4 2 8 4 2 Generalized Pascal triangles Manon Stipulanti (ULiège) 28

Definition: Set of segments of different slopes c : (x, y) (x/2, y/2) (homothety of center (, ), ratio /2) h : (x, y) (x, 2y) A n := i n j i h j (c i (A )) Generalized Pascal triangles Manon Stipulanti (ULiège) 29

Definition: Set of segments of different slopes c : (x, y) (x/2, y/2) (homothety of center (, ), ratio /2) h : (x, y) (x, 2y) A n := i n j i h j (c i (A )) Lemma: (A n ) n is a Cauchy sequence Generalized Pascal triangles Manon Stipulanti (ULiège) 29

Definition: Set of segments of different slopes c : (x, y) (x/2, y/2) (homothety of center (, ), ratio /2) h : (x, y) (x, 2y) A n := i n j i h j (c i (A )) Lemma: (A n ) n is a Cauchy sequence Definition: Limit object L Generalized Pascal triangles Manon Stipulanti (ULiège) 29

A key result Simple characterization of L: topological closure of a union of segments described through a simple combinatorial property Generalized Pascal triangles Manon Stipulanti (ULiège) 3 Theorem (Leroy, Rigo, S., 26) The sequence (U n ) n of compact sets converges to the compact set L when n tends to infinity (for the Hausdorff distance).

Extension modulo p Simplicity: coloring the cells of the grids regarding their parity Extension using Lucas theorem Everything still holds for binomial coefficients r mod p with base-2 expansions of integers p a prime r {,..., p } Theorem (Lucas, 878) Let p be a prime number. If m = m k p k + + m p + m and n = n k p k + + n p + n, then ( ) m k ( ) mi mod p. n i= n i Generalized Pascal triangles Manon Stipulanti (ULiège) 3

Example with p = 3, r = 2 Left: binomial coefficients 2 mod 3 Right: estimate of the corresponding limit object Generalized Pascal triangles Manon Stipulanti (ULiège) 32

Part II: Counting Subword Occurrences

Counting Subword Occurrences Generalized Pascal triangle in base 2 ( u v) v ε n S 2 (n) ε 2 2 3 u 2 3 3 2 4 4 2 5 5 2 2 6 5 3 3 7 4 Definition: S 2 (n) = # { m N ( ) } rep2 (n) > rep 2 (m) n = # (scattered) subwords in {ε} {, } of rep 2 (n) Generalized Pascal triangles Manon Stipulanti (ULiège) 34

The sequence (S 2 (n)) n in the interval [, 256] Palindromic structure regularity Generalized Pascal triangles Manon Stipulanti (ULiège) 35

Regularity in the sense of Allouche and Shallit (992) 2-kernel of s = (s(n)) n K 2 (s) ={(s(n)) n, (s(2n)) n, (s(2n + )) n, (s(4n)) n, (s(4n + )) n, (s(4n + 2)) n,...} ={(s(2 i n + j)) n i and j < 2 i } 2-regular if there exist (t (n)) n,..., (t l (n)) n s.t. each (t(n)) n K 2 (s) is a Z-linear combination of the t j s Generalized Pascal triangles Manon Stipulanti (ULiège) 36

The case of (S 2 (n)) n Theorem (Leroy, Rigo, S., 27) The sequence (S 2 (n)) n satisfies, for all n, S 2 (2n + ) = 3 S 2 (n) S 2 (2n) S 2 (4n) = S 2 (n) + 2 S 2 (2n) S 2 (4n + 2) = 4 S 2 (n) S 2 (2n). Proof using a special tree structure... Generalized Pascal triangles Manon Stipulanti (ULiège) 37

Corollary (Leroy, Rigo, S., 27) (S 2 (n)) n is 2-regular. Proof: Generators (S 2 (n)) n and (S 2 (2n)) n S 2 (n) = S 2 (n) S 2 (2n) = S 2 (2n) S 2 (2n + ) = 3S 2 (n) S 2 (2n) S 2 (4n) = S 2 (n) + 2S 2 (2n) S 2 (4n + ) = S 2 (2(2n) + ) = 3S 2 (2n) S 2 (4n) = 3S 2 (2n) ( S 2 (n) + 2S 2 (2n)) = S 2 (n) + S 2 (2n) S 2 (4n + 2) = 4S 2 (n) 2S 2 (2n) S 2 (4n + 3) = S 2 (2(2n + ) + ) = 3S 2 (2n + ) S 2 (4n + 2) etc. = 3 3S 2 (n) 3S 2 (2n) (4S 2 (n) 2S 2 (2n)) = 5S 2 (n) S 2 (2n) Generalized Pascal triangles Manon Stipulanti (ULiège) 38

Matrix representation to compute (S 2 (n)) n easily ( ) S2 (n) V (n) = S 2 (2n) ( ) ( ) ( S2 (2n) S2 (n) V (2n) = = S 2 (4n) 2 S 2 (2n) }{{} V (2n + ) = ( S2 (2n + ) S 2 (4n + 2) :=µ() ) ( ) 3 = 4 }{{} :=µ() If rep 2 (m) = m l m with m i {, }, then S 2 (m) = ( ) ( ) µ(m ) µ(m l ) ) ( S2 (n) S 2 (2n) ) Generalized Pascal triangles Manon Stipulanti (ULiège) 39

A special tree structure: the tree of subwords T () Generalized Pascal triangles Manon Stipulanti (ULiège) 4 Definition: w a word in {, } The tree of subwords of w is the tree T (w) root ε if u and ua are subwords of w with a {, }, then ua is a child of u Example: w = test test test test test test

Convention in base 2: no leading delete the right part of the tree T (w) new tree T 2 (w) Example: w = length subwords ε 2, 3, 4 T 2 () Generalized Pascal triangles Manon Stipulanti (ULiège) 4

Usefulness: #nodes on level n of T 2 (w) =#subwords of length n of w in {, } {ε} Generalized Pascal triangles Manon Stipulanti (ULiège) 42

Usefulness: #nodes on level n of T 2 (w) =#subwords of length n of w Example: w = in {, } {ε} level #nodes subwords ε 2 2, 3 2, 4 Total 7 T 2 () Link with S 2 : val 2 () = 9 S 2 (9) = 7 Generalized Pascal triangles Manon Stipulanti (ULiège) 42

An example of a proof using T 2 Palindromic structure (Leroy, Rigo, S., 27) For all l > and all r < 2 l, S 2 (2 l + r) = S 2 (2 l+ r ). Example: l = 3, r = 2 3 + = 9 rep 2 (9) = 2 4 = 4 rep 2 (4) = T 2() T 2() T 2 () = T 2 () # nodes of T 2 () = # nodes of T 2 () S 2 (9) = S 2 (4) Generalized Pascal triangles Manon Stipulanti (ULiège) 43

Link with the Farey tree Definition: The Farey tree is a tree that contains every (reduced) positive rational less than exactly once. 2 3 4 5 2 7 2 5 3 8 3 7 2 3 3 5 4 7 5 8 3 4 5 7 4 5 S 2(n) 2 3 3 4 5 5 4 5 7 8 7 7 8 7 5... Generalized Pascal triangles Manon Stipulanti (ULiège) 44

Part III: Behavior of the Summatory Function

Example: s(n) number of in rep 2 (n) s(2n) = s(n) s(2n+) = s(n)+ s is 2-regular Summatory function A: A() := n A(n) := s(j) n j= Theorem (Delange, 975) A(n) n = 2 log 2(n) + G(log 2 (n)) () where G continuous, nowhere differentiable, periodic of period. Generalized Pascal triangles Manon Stipulanti (ULiège) 46

Theorem (Allouche, Shallit, 23) Under some hypotheses, the summatory function of every b- regular sequence has a behavior analogous to (). Replacing s by S 2 : same behavior as () but does not satisfy the hypotheses of the theorem Generalized Pascal triangles Manon Stipulanti (ULiège) 47

Using several numeration systems Definition: A 2 () := n A 2 (n) := S 2 (j) n j= First few values:,, 3, 6, 9, 3, 8, 23, 27, 32, 39, 47, 54, 6, 69, 76, 8, 87, 96, 7,... Generalized Pascal triangles Manon Stipulanti (ULiège) 48

Using several numeration systems Definition: A 2 () := n A 2 (n) := S 2 (j) n j= First few values:,, 3, 6, 9, 3, 8, 23, 27, 32, 39, 47, 54, 6, 69, 76, 8, 87, 96, 7,... Lemma (Leroy, Rigo, S., 27) For all n, A 2 (2 n ) = 3 n. Generalized Pascal triangles Manon Stipulanti (ULiège) 48

Lemma (Leroy, Rigo, S., 27) Let l. If r 2 l, then A 2 (2 l + r) = 2 3 l + A 2 (2 l + r) + A 2 (r). If 2 l < r < 2 l, then A 2 (2 l +r) = 4 3 l 2 3 l A 2 (2 l +r ) A 2 (r ) where r = 2 l r. Generalized Pascal triangles Manon Stipulanti (ULiège) 49

Lemma (Leroy, Rigo, S., 27) Let l. If r 2 l, then A 2 (2 l + r) = 2 3 l + A 2 (2 l + r) + A 2 (r). If 2 l < r < 2 l, then A 2 (2 l +r) = 4 3 l 2 3 l A 2 (2 l +r ) A 2 (r ) where r = 2 l r. 3-decomposition: particular decomposition of A 2 (n) based on powers of 3 Generalized Pascal triangles Manon Stipulanti (ULiège) 49

Lemma (Leroy, Rigo, S., 27) Let l. If r 2 l, then A 2 (2 l + r) = 2 3 l + A 2 (2 l + r) + A 2 (r). If 2 l < r < 2 l, then A 2 (2 l +r) = 4 3 l 2 3 l A 2 (2 l +r ) A 2 (r ) where r = 2 l r. 3-decomposition: particular decomposition of A 2 (n) based on powers of 3 two numeration systems: base 2 and base 3 Generalized Pascal triangles Manon Stipulanti (ULiège) 49

Theorem (Leroy, Rigo, S., 27) There exists a continuous and periodic function H 2 of period such that, for all n, A 2 (n) = 3 log 2 (n) H 2 (log 2 (n)). Generalized Pascal triangles Manon Stipulanti (ULiège) 5

Conclusion In this talk: Generalization of the Pascal triangle in base 2 modulo a prime number 2-regularity of the sequence (S 2 (n)) n counting subword occurrences Behavior of the summatory function (A 2 (n)) n of the sequence (S 2 (n)) n Generalized Pascal triangles Manon Stipulanti (ULiège) 5

Other work Done: Generalization of the Pascal triangle modulo a prime number: extension to any Pisot Bertrand numeration system Regularity of the sequence counting subword occurrences: extension to any base b and the Fibonacci numeration system Behavior of the summatory function: extension to any base b (exact behavior) and the Fibonacci numeration system (asymptotics) What s next? Pisot Bertrand numeration systems, Apply the methods for sequences not related to Pascal triangles, etc. Generalized Pascal triangles Manon Stipulanti (ULiège) 52

Conus textile or Cloth of gold cone Color pattern of its shell Sierpiński gasket Generalized Pascal triangles Manon Stipulanti (ULiège) 53

References J.-P. Allouche, J. Shallit, The ring of k-regular sequences, Theoret. Comput. Sci. 98(2) (992), 63 97. J.-P. Allouche, J. Shallit, Automatic sequences. Cambridge University Press, Cambridge, 23. Theory, applications, generalizations, H. Delange, Sur la fonction sommatoire de la fonction somme des chiffres, Enseignement Math. 2 (975), 3 47. P. Dumas, Diviser pour régner : algèbre et analyse, Journées Aléa 26 (CIRM). F. von Haeseler, H.-O. Peitgen, G. Skordev, Pascal s triangle, dynamical systems and attractors, Ergod. Th. & Dynam. Sys. 2 (992), 479 486. J. Leroy, M. Rigo, M. Stipulanti, Generalized Pascal triangle for binomial coefficients of words, Adv. in Appl. Math. 8 (26), 24 47. J. Leroy, M. Rigo, M. Stipulanti, Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Math. 34 (27) 862 88. J. Leroy, M. Rigo, M. Stipulanti, Behavior of digital sequences through exotic numeration systems, Electron. J. Combin. 24 (27), no., Paper.44, 36 pp. J. Leroy, M. Rigo, M. Stipulanti, Counting Subword Occurrences in Base-b Expansions, submitted in April 27. M. Lothaire, Combinatorics On Words, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 997. É. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math. (878) 97 24. M. Stipulanti, Convergence of Pascal-Like Triangles in Pisot Bertrand Numeration Systems, in progress. Generalized Pascal triangles Manon Stipulanti (ULiège) 54