Extremal generalized smooth words Kolakoski word Run-length - - PowerPoint PPT Presentation

▶
extremal generalized smooth words
SMART_READER_LITE
LIVE PREVIEW

Extremal generalized smooth words Kolakoski word Run-length - - PowerPoint PPT Presentation

Mar 14, 2024 420 likes •1.1k views

Extremal generalized smooth words . . . G. Paquin, S. Brlek, D. Jamet Outlines Introduction Extremal generalized smooth words Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems eve Paquin (1) cko Brlek

slide-1
SLIDE 1

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Extremal generalized smooth words

Genevi` eve Paquin(1) Sreˇ cko Brlek(1) Damien Jamet(2)

(1) LaCIM - Laboratoire de Combinatoire et d’Informatique Math´ ematique UQAM - Universit´ e du Qu´ ebec ` a Montr´ eal (2) LORIA - INRIA Lorraine web : http://www.labmath.uqam.ca/˜ paquin/ email : [paquin,brlek]@lacim.uqam.ca, jamet@lirmm.fr

September 2, 2006

slide-2
SLIDE 2

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

slide-3
SLIDE 3

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-4
SLIDE 4

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-5
SLIDE 5

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-6
SLIDE 6

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-7
SLIDE 7

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-8
SLIDE 8

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-9
SLIDE 9

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-10
SLIDE 10

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-11
SLIDE 11

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-12
SLIDE 12

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-13
SLIDE 13

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-14
SLIDE 14

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-15
SLIDE 15

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-16
SLIDE 16

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-17
SLIDE 17

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Kolakoski word

◮ Kolakoski word (Kolakoski W., 1965) : Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965).

K = 22112122122112112212112122112112122122112122121121122 · · · = 22112122122112112212112122112112122122112122121121122 · · ·

◮ (Dekking F. M., 1980-1981) : Dekking F.M., On the structure of self generating sequences, S´

  • em. de th´

eorie des nombres de Bordeaux (1980-1981). ◮ Density of the letters in K ? ◮ Recurrence ? ◮ Closure under reversal ? ◮ (Weakley W. D., 1989) : Weakley W. D., On the number of C∞-words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989).

The complexity function of K is in O(n

log 3 log 2 ). ◮ (Carpi A., 1994) : Carpi A., On repeated factors in C∞-words, Information Processing Letters 52, p.289-294, (1994). ◮ K does contain only a finite number of squares ◮ K is cube-free

slide-18
SLIDE 18

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Run-length encoding

◮ Every word w ∈ {1, 2}ω can be uniquely written as w = 1i12i21i3 · · · ,

ik ≥ 0 .

◮ Example : w = 1221112121112 · · · = 11221321112113 · · · ◮ We define ∆ : Σω −

→ Nω, by ∆(w) = i1, i2, i3, · · ·

◮ Example : ∆(w) = [1, 2, 3, 1, 1, 1, 3, · · · ] which is usually written as

∆(w) = 1231113 · · ·

◮ The operator ∆ may be iterated until the coding alphabet changes. ◮ Example : Let u = 1121122121121221121121221211221221121121 · · ·

∆0(u) = 1121122121121221121121221211221221121121 · · · ∆1(u) = 21221121122121121122122121 · · · ∆2(u) = 11221221121221211 · · · ∆3(u) = 2212211211 · · · ∆4(u) = 21221 · · ·

slide-19
SLIDE 19

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Run-length encoding

◮ Every word w ∈ {1, 2}ω can be uniquely written as w = 1i12i21i3 · · · ,

ik ≥ 0 .

◮ Example : w = 1221112121112 · · · = 11221321112113 · · · ◮ We define ∆ : Σω −

→ Nω, by ∆(w) = i1, i2, i3, · · ·

◮ Example : ∆(w) = [1, 2, 3, 1, 1, 1, 3, · · · ] which is usually written as

∆(w) = 1231113 · · ·

◮ The operator ∆ may be iterated until the coding alphabet changes. ◮ Example : Let u = 1121122121121221121121221211221221121121 · · ·

∆0(u) = 1121122121121221121121221211221221121121 · · · ∆1(u) = 21221121122121121122122121 · · · ∆2(u) = 11221221121221211 · · · ∆3(u) = 2212211211 · · · ∆4(u) = 21221 · · ·

slide-20
SLIDE 20

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Run-length encoding

◮ Every word w ∈ {1, 2}ω can be uniquely written as w = 1i12i21i3 · · · ,

ik ≥ 0 .

◮ Example : w = 1221112121112 · · · = 11221321112113 · · · ◮ We define ∆ : Σω −

→ Nω, by ∆(w) = i1, i2, i3, · · ·

◮ Example : ∆(w) = [1, 2, 3, 1, 1, 1, 3, · · · ] which is usually written as

∆(w) = 1231113 · · ·

◮ The operator ∆ may be iterated until the coding alphabet changes. ◮ Example : Let u = 1121122121121221121121221211221221121121 · · ·

∆0(u) = 1121122121121221121121221211221221121121 · · · ∆1(u) = 21221121122121121122122121 · · · ∆2(u) = 11221221121221211 · · · ∆3(u) = 2212211211 · · · ∆4(u) = 21221 · · ·

slide-21
SLIDE 21

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Run-length encoding

◮ Every word w ∈ {1, 2}ω can be uniquely written as w = 1i12i21i3 · · · ,

ik ≥ 0 .

◮ Example : w = 1221112121112 · · · = 11221321112113 · · · ◮ We define ∆ : Σω −

→ Nω, by ∆(w) = i1, i2, i3, · · ·

◮ Example : ∆(w) = [1, 2, 3, 1, 1, 1, 3, · · · ] which is usually written as

∆(w) = 1231113 · · ·

◮ The operator ∆ may be iterated until the coding alphabet changes. ◮ Example : Let u = 1121122121121221121121221211221221121121 · · ·

∆0(u) = 1121122121121221121121221211221221121121 · · · ∆1(u) = 21221121122121121122122121 · · · ∆2(u) = 11221221121221211 · · · ∆3(u) = 2212211211 · · · ∆4(u) = 21221 · · ·

slide-22
SLIDE 22

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Run-length encoding

◮ Every word w ∈ {1, 2}ω can be uniquely written as w = 1i12i21i3 · · · ,

ik ≥ 0 .

◮ Example : w = 1221112121112 · · · = 11221321112113 · · · ◮ We define ∆ : Σω −

→ Nω, by ∆(w) = i1, i2, i3, · · ·

◮ Example : ∆(w) = [1, 2, 3, 1, 1, 1, 3, · · · ] which is usually written as

∆(w) = 1231113 · · ·

◮ The operator ∆ may be iterated until the coding alphabet changes. ◮ Example : Let u = 1121122121121221121121221211221221121121 · · ·

∆0(u) = 1121122121121221121121221211221221121121 · · · ∆1(u) = 21221121122121121122122121 · · · ∆2(u) = 11221221121221211 · · · ∆3(u) = 2212211211 · · · ∆4(u) = 21221 · · ·

slide-23
SLIDE 23

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Run-length encoding

◮ Every word w ∈ {1, 2}ω can be uniquely written as w = 1i12i21i3 · · · ,

ik ≥ 0 .

◮ Example : w = 1221112121112 · · · = 11221321112113 · · · ◮ We define ∆ : Σω −

→ Nω, by ∆(w) = i1, i2, i3, · · ·

◮ Example : ∆(w) = [1, 2, 3, 1, 1, 1, 3, · · · ] which is usually written as

∆(w) = 1231113 · · ·

◮ The operator ∆ may be iterated until the coding alphabet changes. ◮ Example : Let u = 1121122121121221121121221211221221121121 · · ·

∆0(u) = 1121122121121221121121221211221221121121 · · · ∆1(u) = 21221121122121121122122121 · · · ∆2(u) = 11221221121221211 · · · ∆3(u) = 2212211211 · · · ∆4(u) = 21221 · · ·

slide-24
SLIDE 24

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Smooth words

◮ The set of smooth words over Σ = {1, 2} is defined as

KΣ = {w ∈ Σω | ∀k ∈ N, ∆k(w) ∈ Σω}.

Brlek S., Ladouceur A., A note on differentiable palindromes, Theoretical Computer Science 302, p.167-178 (2003). ◮ It can be generalized to any 2-letter ordered alphabet Σ = {a, b}. Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. comp. sc. 341, p. 293-310 (2005). ◮ Example : over the alphabet Σ = {1, 3}, v = 1113111333131 · · · is not

smooth, since ∆(v) = 313311 · · · and ∆2(v) = 112 · · · .

◮ The bijection Φ : KΣ → Σω is defined by

Φ(w)[j + 1] = ∆j(w)[1] for j ≥ 0

Lamas P, Contribution ` a l’´ etude de quelques mots infinis (1995) Dekking F.M., What is the long range order in the Kolakoski sequence (1997). ◮ Example : if u = 112112212112122112112122121122 · · · , then

Φ(u) = 12122 · · ·

slide-25
SLIDE 25

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Smooth words

◮ The set of smooth words over Σ = {1, 2} is defined as

KΣ = {w ∈ Σω | ∀k ∈ N, ∆k(w) ∈ Σω}.

Brlek S., Ladouceur A., A note on differentiable palindromes, Theoretical Computer Science 302, p.167-178 (2003). ◮ It can be generalized to any 2-letter ordered alphabet Σ = {a, b}. Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. comp. sc. 341, p. 293-310 (2005). ◮ Example : over the alphabet Σ = {1, 3}, v = 1113111333131 · · · is not

smooth, since ∆(v) = 313311 · · · and ∆2(v) = 112 · · · .

◮ The bijection Φ : KΣ → Σω is defined by

Φ(w)[j + 1] = ∆j(w)[1] for j ≥ 0

Lamas P, Contribution ` a l’´ etude de quelques mots infinis (1995) Dekking F.M., What is the long range order in the Kolakoski sequence (1997). ◮ Example : if u = 112112212112122112112122121122 · · · , then

Φ(u) = 12122 · · ·

slide-26
SLIDE 26

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Smooth words

◮ The set of smooth words over Σ = {1, 2} is defined as

KΣ = {w ∈ Σω | ∀k ∈ N, ∆k(w) ∈ Σω}.

Brlek S., Ladouceur A., A note on differentiable palindromes, Theoretical Computer Science 302, p.167-178 (2003). ◮ It can be generalized to any 2-letter ordered alphabet Σ = {a, b}. Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. comp. sc. 341, p. 293-310 (2005). ◮ Example : over the alphabet Σ = {1, 3}, v = 1113111333131 · · · is not

smooth, since ∆(v) = 313311 · · · and ∆2(v) = 112 · · · .

◮ The bijection Φ : KΣ → Σω is defined by

Φ(w)[j + 1] = ∆j(w)[1] for j ≥ 0

Lamas P, Contribution ` a l’´ etude de quelques mots infinis (1995) Dekking F.M., What is the long range order in the Kolakoski sequence (1997). ◮ Example : if u = 112112212112122112112122121122 · · · , then

Φ(u) = 12122 · · ·

slide-27
SLIDE 27

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Smooth words

◮ The set of smooth words over Σ = {1, 2} is defined as

KΣ = {w ∈ Σω | ∀k ∈ N, ∆k(w) ∈ Σω}.

Brlek S., Ladouceur A., A note on differentiable palindromes, Theoretical Computer Science 302, p.167-178 (2003). ◮ It can be generalized to any 2-letter ordered alphabet Σ = {a, b}. Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. comp. sc. 341, p. 293-310 (2005). ◮ Example : over the alphabet Σ = {1, 3}, v = 1113111333131 · · · is not

smooth, since ∆(v) = 313311 · · · and ∆2(v) = 112 · · · .

◮ The bijection Φ : KΣ → Σω is defined by

Φ(w)[j + 1] = ∆j(w)[1] for j ≥ 0

Lamas P, Contribution ` a l’´ etude de quelques mots infinis (1995) Dekking F.M., What is the long range order in the Kolakoski sequence (1997). ◮ Example : if u = 112112212112122112112122121122 · · · , then

Φ(u) = 12122 · · ·

slide-28
SLIDE 28

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Smooth words

◮ The set of smooth words over Σ = {1, 2} is defined as

KΣ = {w ∈ Σω | ∀k ∈ N, ∆k(w) ∈ Σω}.

Brlek S., Ladouceur A., A note on differentiable palindromes, Theoretical Computer Science 302, p.167-178 (2003). ◮ It can be generalized to any 2-letter ordered alphabet Σ = {a, b}. Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. comp. sc. 341, p. 293-310 (2005). ◮ Example : over the alphabet Σ = {1, 3}, v = 1113111333131 · · · is not

smooth, since ∆(v) = 313311 · · · and ∆2(v) = 112 · · · .

◮ The bijection Φ : KΣ → Σω is defined by

Φ(w)[j + 1] = ∆j(w)[1] for j ≥ 0

Lamas P, Contribution ` a l’´ etude de quelques mots infinis (1995) Dekking F.M., What is the long range order in the Kolakoski sequence (1997). ◮ Example : if u = 112112212112122112112122121122 · · · , then

Φ(u) = 12122 · · ·

slide-29
SLIDE 29

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Generalized Kolakoski words

◮ In K{1,2}, the operator ∆ has two fixpoints, namely

∆(K) = K, ∆(1K) = 1K, where K is the Kolakoski word.

◮ Over a 2-letter ordered alphabet Σ = {a, b}, there are two generalized

Kolakoski words, K(a,b) and K(b,a), where K(a,b) (resp. K(b,a)) begins by a (resp. b) and satisfies ∆(K(a,b)) = K(a,b) (resp. ∆(K(b,a) = K(b,a)).

◮ For instance, K(2,1) = K and K(1,2) = 1K ◮ Examples :

K(1,3) = 1 |{z}

1

333 |{z}

3

111 |{z}

3

333 |{z}

3

1 |{z}

1

3 |{z}

1

1 |{z}

1

333 |{z}

3

111 |{z}

3

333 |{z}

3

1 |{z}

1

333 |{z}

3

1 |{z}

1

· · · K(3,2) = 333 |{z}

3

222 |{z}

3

333 |{z}

3

22 |{z}

2

33 |{z}

2

22 |{z}

2

333 |{z}

3

222 |{z}

3

333 |{z}

3

22 |{z}

2

33 |{z}

2

222 |{z}

3

333 |{z}

3

· · ·

slide-30
SLIDE 30

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Generalized Kolakoski words

◮ In K{1,2}, the operator ∆ has two fixpoints, namely

∆(K) = K, ∆(1K) = 1K, where K is the Kolakoski word.

◮ Over a 2-letter ordered alphabet Σ = {a, b}, there are two generalized

Kolakoski words, K(a,b) and K(b,a), where K(a,b) (resp. K(b,a)) begins by a (resp. b) and satisfies ∆(K(a,b)) = K(a,b) (resp. ∆(K(b,a) = K(b,a)).

◮ For instance, K(2,1) = K and K(1,2) = 1K ◮ Examples :

K(1,3) = 1 |{z}

1

333 |{z}

3

111 |{z}

3

333 |{z}

3

1 |{z}

1

3 |{z}

1

1 |{z}

1

333 |{z}

3

111 |{z}

3

333 |{z}

3

1 |{z}

1

333 |{z}

3

1 |{z}

1

· · · K(3,2) = 333 |{z}

3

222 |{z}

3

333 |{z}

3

22 |{z}

2

33 |{z}

2

22 |{z}

2

333 |{z}

3

222 |{z}

3

333 |{z}

3

22 |{z}

2

33 |{z}

2

222 |{z}

3

333 |{z}

3

· · ·

slide-31
SLIDE 31

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Generalized Kolakoski words

◮ In K{1,2}, the operator ∆ has two fixpoints, namely

∆(K) = K, ∆(1K) = 1K, where K is the Kolakoski word.

◮ Over a 2-letter ordered alphabet Σ = {a, b}, there are two generalized

Kolakoski words, K(a,b) and K(b,a), where K(a,b) (resp. K(b,a)) begins by a (resp. b) and satisfies ∆(K(a,b)) = K(a,b) (resp. ∆(K(b,a) = K(b,a)).

◮ For instance, K(2,1) = K and K(1,2) = 1K ◮ Examples :

K(1,3) = 1 |{z}

1

333 |{z}

3

111 |{z}

3

333 |{z}

3

1 |{z}

1

3 |{z}

1

1 |{z}

1

333 |{z}

3

111 |{z}

3

333 |{z}

3

1 |{z}

1

333 |{z}

3

1 |{z}

1

· · · K(3,2) = 333 |{z}

3

222 |{z}

3

333 |{z}

3

22 |{z}

2

33 |{z}

2

22 |{z}

2

333 |{z}

3

222 |{z}

3

333 |{z}

3

22 |{z}

2

33 |{z}

2

222 |{z}

3

333 |{z}

3

· · ·

slide-32
SLIDE 32

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Generalized Kolakoski words

◮ In K{1,2}, the operator ∆ has two fixpoints, namely

∆(K) = K, ∆(1K) = 1K, where K is the Kolakoski word.

◮ Over a 2-letter ordered alphabet Σ = {a, b}, there are two generalized

Kolakoski words, K(a,b) and K(b,a), where K(a,b) (resp. K(b,a)) begins by a (resp. b) and satisfies ∆(K(a,b)) = K(a,b) (resp. ∆(K(b,a) = K(b,a)).

◮ For instance, K(2,1) = K and K(1,2) = 1K ◮ Examples :

K(1,3) = 1 |{z}

1

333 |{z}

3

111 |{z}

3

333 |{z}

3

1 |{z}

1

3 |{z}

1

1 |{z}

1

333 |{z}

3

111 |{z}

3

333 |{z}

3

1 |{z}

1

333 |{z}

3

1 |{z}

1

· · · K(3,2) = 333 |{z}

3

222 |{z}

3

333 |{z}

3

22 |{z}

2

33 |{z}

2

22 |{z}

2

333 |{z}

3

222 |{z}

3

333 |{z}

3

22 |{z}

2

33 |{z}

2

222 |{z}

3

333 |{z}

3

· · ·

slide-33
SLIDE 33

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Problems

◮ Are generalized Kolakoski words recurrent ? closed under reversal ? What

are the densities of the letters ?

◮ What are the extremal smooth words over a 2-letter alphabets

Σ = {a, b} : minimal m{a,b} and maximal M{a,b} w.r.t. lexicographic

  • rder ?

Remark that for 2-letter alphabets, M = m.

slide-34
SLIDE 34

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Problems

◮ Are generalized Kolakoski words recurrent ? closed under reversal ? What

are the densities of the letters ?

◮ What are the extremal smooth words over a 2-letter alphabets

Σ = {a, b} : minimal m{a,b} and maximal M{a,b} w.r.t. lexicographic

  • rder ?

Remark that for 2-letter alphabets, M = m.

slide-35
SLIDE 35

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 2}

Computation of extremal smooth words over {1, 2}

m{1,2} = 1121122121121221121121221211221221121121221121122121 · · · M{1,2} = 2212211212212112212212112122112112212212112212211212 · · · Naive algorithm : complexity O(n2 log(n)).

Characterization of m{1,2}

◮ Φ(m{1,2}) = 12122121122211211121122211112212111122221211 · · · ◮ m{1,2} is not a Lyndon word ◮ Conjecture : m{1,2} has a finite Lyndon factorization ◮ No efficient algorithm known Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-36
SLIDE 36

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 2}

Computation of extremal smooth words over {1, 2}

m{1,2} = 1121122121121221121121221211221221121121221121122121 · · · M{1,2} = 2212211212212112212212112122112112212212112212211212 · · · Naive algorithm : complexity O(n2 log(n)).

Characterization of m{1,2}

◮ Φ(m{1,2}) = 12122121122211211121122211112212111122221211 · · · ◮ m{1,2} is not a Lyndon word ◮ Conjecture : m{1,2} has a finite Lyndon factorization ◮ No efficient algorithm known Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-37
SLIDE 37

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 2}

Computation of extremal smooth words over {1, 2}

m{1,2} = 1121122121121221121121221211221221121121221121122121 · · · M{1,2} = 2212211212212112212212112122112112212212112212211212 · · · Naive algorithm : complexity O(n2 log(n)).

Characterization of m{1,2}

◮ Φ(m{1,2}) = 12122121122211211121122211112212111122221211 · · · ◮ m{1,2} is not a Lyndon word ◮ Conjecture : m{1,2} has a finite Lyndon factorization ◮ No efficient algorithm known Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-38
SLIDE 38

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 2}

Computation of extremal smooth words over {1, 2}

m{1,2} = 1121122121121221121121221211221221121121221121122121 · · · M{1,2} = 2212211212212112212212112122112112212212112212211212 · · · Naive algorithm : complexity O(n2 log(n)).

Characterization of m{1,2}

◮ Φ(m{1,2}) = 12122121122211211121122211112212111122221211 · · · ◮ m{1,2} is not a Lyndon word ◮ Conjecture : m{1,2} has a finite Lyndon factorization ◮ No efficient algorithm known Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-39
SLIDE 39

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 2}

Computation of extremal smooth words over {1, 2}

m{1,2} = 1121122121121221121121221211221221121121221121122121 · · · M{1,2} = 2212211212212112212212112122112112212212112212211212 · · · Naive algorithm : complexity O(n2 log(n)).

Characterization of m{1,2}

◮ Φ(m{1,2}) = 12122121122211211121122211112212111122221211 · · · ◮ m{1,2} is not a Lyndon word ◮ Conjecture : m{1,2} has a finite Lyndon factorization ◮ No efficient algorithm known Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-40
SLIDE 40

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 2}

Computation of extremal smooth words over {1, 2}

m{1,2} = 1121122121121221121121221211221221121121221121122121 · · · M{1,2} = 2212211212212112212212112122112112212212112212211212 · · · Naive algorithm : complexity O(n2 log(n)).

Characterization of m{1,2}

◮ Φ(m{1,2}) = 12122121122211211121122211112212111122221211 · · · ◮ m{1,2} is not a Lyndon word ◮ Conjecture : m{1,2} has a finite Lyndon factorization ◮ No efficient algorithm known Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-41
SLIDE 41

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 3}

Minimal smooth word over {1, 3}

◮ m{1,3} = 11131113131113111313111313111311131311131113131113 · · · ◮ Φ(m{1,3}) = (13)ω ◮ Linear algorithm to compute m{1,3} ◮ Recall : Φ(F) = 112(13)ω, F the infinite Fibonacci word Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. Comp. Sc., 341, p. 293-310 (2005). ◮ m{1,3} = ∆3(F) ◮ Recursive definition for m{1,3} Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-42
SLIDE 42

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 3}

Minimal smooth word over {1, 3}

◮ m{1,3} = 11131113131113111313111313111311131311131113131113 · · · ◮ Φ(m{1,3}) = (13)ω ◮ Linear algorithm to compute m{1,3} ◮ Recall : Φ(F) = 112(13)ω, F the infinite Fibonacci word Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. Comp. Sc., 341, p. 293-310 (2005). ◮ m{1,3} = ∆3(F) ◮ Recursive definition for m{1,3} Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-43
SLIDE 43

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 3}

Minimal smooth word over {1, 3}

◮ m{1,3} = 11131113131113111313111313111311131311131113131113 · · · ◮ Φ(m{1,3}) = (13)ω ◮ Linear algorithm to compute m{1,3} ◮ Recall : Φ(F) = 112(13)ω, F the infinite Fibonacci word Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. Comp. Sc., 341, p. 293-310 (2005). ◮ m{1,3} = ∆3(F) ◮ Recursive definition for m{1,3} Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-44
SLIDE 44

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 3}

Minimal smooth word over {1, 3}

◮ m{1,3} = 11131113131113111313111313111311131311131113131113 · · · ◮ Φ(m{1,3}) = (13)ω ◮ Linear algorithm to compute m{1,3} ◮ Recall : Φ(F) = 112(13)ω, F the infinite Fibonacci word Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. Comp. Sc., 341, p. 293-310 (2005). ◮ m{1,3} = ∆3(F) ◮ Recursive definition for m{1,3} Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-45
SLIDE 45

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 3}

Minimal smooth word over {1, 3}

◮ m{1,3} = 11131113131113111313111313111311131311131113131113 · · · ◮ Φ(m{1,3}) = (13)ω ◮ Linear algorithm to compute m{1,3} ◮ Recall : Φ(F) = 112(13)ω, F the infinite Fibonacci word Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. Comp. Sc., 341, p. 293-310 (2005). ◮ m{1,3} = ∆3(F) ◮ Recursive definition for m{1,3} Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-46
SLIDE 46

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Over Σ = {1, 3}

Minimal smooth word over {1, 3}

◮ m{1,3} = 11131113131113111313111313111311131311131113131113 · · · ◮ Φ(m{1,3}) = (13)ω ◮ Linear algorithm to compute m{1,3} ◮ Recall : Φ(F) = 112(13)ω, F the infinite Fibonacci word Berth´ e V., Brlek S., Choquette P., Smooth words over arbitrary alphabets, Theor. Comp. Sc., 341, p. 293-310 (2005). ◮ m{1,3} = ∆3(F) ◮ Recursive definition for m{1,3} Paquin G., Melan¸ con G., Brlek S., Properties of the extremal infinite smooth words, Journal of Automata, Langagues and Combinatorics, to appear (accepted in 2005).

slide-47
SLIDE 47

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Periodicity

Odd alphabet

◮ Φ(m{a,b}) = (ab)ω ◮ Linear algorithm :

a b I

ε/ab a/(bbab)

a−1 2

α/(abbb)

α−1 2 ab

α/(baaa)

α−1 2 ba

Even alphabet

◮ Φ(m{a,b}) = abω ◮ Linear algorithm :

I b

aa/abb αα/aαbα ε/ab−1

◮ Remark : M{a,b}= K(b,a).

slide-48
SLIDE 48

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Examples

m{3,5} = 3333355555333335553335553333355555333335553335553 · · · ∆(m{3,5}) = 5553335553335553333355555333335555533333555333555 · · · ∆2(m{3,5}) = 3333355555333335553335553333355555333335553335553 · · · ∆3(m{3,5}) = 5553335553335553333355555333335555533333555333555 · · · ∆4(m{3,5}) = 333335555533333 · · · ∆5(m{3,5}) = 555 · · · ∆6(m{3,5}) = 3 · · · m{2,4} = 222244442222444422442244222244442222444422442244 · · · ∆(m{2,4}) = 444422224444222244224422444422224444222244224422 · · · ∆2(m{2,4}) = 444422224444222244224422444422224444222244224422 · · · ∆3(m{2,4}) = 4444222244442222 · · · ∆4(m{2,4}) = 4444 · · · ∆5(m{2,4}) = 4 · · ·

slide-49
SLIDE 49

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Recurrence

Let u ∈ Σk.

Odd alphabet

◮ Φ−1(u) is a palindrome of

  • dd length.

◮ For every smooth word w

(extremal or not), F(w) is closed under reversal.

◮ Every smooth word

(extremal or not) is recurrent.

Even alphabet

◮ Φ−1(u) has an even length

and contains arbitrary long squares.

◮ The set F(w), w an

extremal word, is NOT closed under reversal.

◮ Every smooth word is

recurrent.

slide-50
SLIDE 50

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Recurrence

Let u ∈ Σk.

Odd alphabet

◮ Φ−1(u) is a palindrome of

  • dd length.

◮ For every smooth word w

(extremal or not), F(w) is closed under reversal.

◮ Every smooth word

(extremal or not) is recurrent.

Even alphabet

◮ Φ−1(u) has an even length

and contains arbitrary long squares.

◮ The set F(w), w an

extremal word, is NOT closed under reversal.

◮ Every smooth word is

recurrent.

slide-51
SLIDE 51

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Recurrence

Let u ∈ Σk.

Odd alphabet

◮ Φ−1(u) is a palindrome of

  • dd length.

◮ For every smooth word w

(extremal or not), F(w) is closed under reversal.

◮ Every smooth word

(extremal or not) is recurrent.

Even alphabet

◮ Φ−1(u) has an even length

and contains arbitrary long squares.

◮ The set F(w), w an

extremal word, is NOT closed under reversal.

◮ Every smooth word is

recurrent.

slide-52
SLIDE 52

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Recurrence

Let u ∈ Σk.

Odd alphabet

◮ Φ−1(u) is a palindrome of

  • dd length.

◮ For every smooth word w

(extremal or not), F(w) is closed under reversal.

◮ Every smooth word

(extremal or not) is recurrent.

Even alphabet

◮ Φ−1(u) has an even length

and contains arbitrary long squares.

◮ The set F(w), w an

extremal word, is NOT closed under reversal.

◮ Every smooth word is

recurrent.

slide-53
SLIDE 53

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Recurrence

Let u ∈ Σk.

Odd alphabet

◮ Φ−1(u) is a palindrome of

  • dd length.

◮ For every smooth word w

(extremal or not), F(w) is closed under reversal.

◮ Every smooth word

(extremal or not) is recurrent.

Even alphabet

◮ Φ−1(u) has an even length

and contains arbitrary long squares.

◮ The set F(w), w an

extremal word, is NOT closed under reversal.

◮ Every smooth word is

recurrent.

slide-54
SLIDE 54

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Recurrence

Let u ∈ Σk.

Odd alphabet

◮ Φ−1(u) is a palindrome of

  • dd length.

◮ For every smooth word w

(extremal or not), F(w) is closed under reversal.

◮ Every smooth word

(extremal or not) is recurrent.

Even alphabet

◮ Φ−1(u) has an even length

and contains arbitrary long squares.

◮ The set F(w), w an

extremal word, is NOT closed under reversal.

◮ Every smooth word is

recurrent.

slide-55
SLIDE 55

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Lyndon factorization

Odd alphabet

◮ m{a,b} is an infinite

Lyndon word if and only if a = 1.

◮ The Lyndon factorization

  • f ∆(m{1,b}) is an infinite

sequence of finite Lyndon words.

Even alphabet

◮ m{a,b} is an infinite

Lyndon word.

◮ The complement of K(b,a)

is an infinite Lyndon word.

slide-56
SLIDE 56

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Lyndon factorization

Odd alphabet

◮ m{a,b} is an infinite

Lyndon word if and only if a = 1.

◮ The Lyndon factorization

  • f ∆(m{1,b}) is an infinite

sequence of finite Lyndon words.

Even alphabet

◮ m{a,b} is an infinite

Lyndon word.

◮ The complement of K(b,a)

is an infinite Lyndon word.

slide-57
SLIDE 57

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Lyndon factorization

Odd alphabet

◮ m{a,b} is an infinite

Lyndon word if and only if a = 1.

◮ The Lyndon factorization

  • f ∆(m{1,b}) is an infinite

sequence of finite Lyndon words.

Even alphabet

◮ m{a,b} is an infinite

Lyndon word.

◮ The complement of K(b,a)

is an infinite Lyndon word.

slide-58
SLIDE 58

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Lyndon factorization

Odd alphabet

◮ m{a,b} is an infinite

Lyndon word if and only if a = 1.

◮ The Lyndon factorization

  • f ∆(m{1,b}) is an infinite

sequence of finite Lyndon words.

Even alphabet

◮ m{a,b} is an infinite

Lyndon word.

◮ The complement of K(b,a)

is an infinite Lyndon word.

slide-59
SLIDE 59

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Density

Odd alphabet

◮ If Σ = {1, b}, the density

  • f the letter b for the

minimal smooth word is given by 1/( √ 2b − 1 − 1)

◮ If Σ = {a, b}, a = 1 : still

an open problem.

Even alphabet

◮ The densities of the letters

a and b are 0.5.

◮ It follows that the densities

  • f the letters in K(a,b) and

K(b,a) are 0.5.

slide-60
SLIDE 60

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Density

Odd alphabet

◮ If Σ = {1, b}, the density

  • f the letter b for the

minimal smooth word is given by 1/( √ 2b − 1 − 1)

◮ If Σ = {a, b}, a = 1 : still

an open problem.

Even alphabet

◮ The densities of the letters

a and b are 0.5.

◮ It follows that the densities

  • f the letters in K(a,b) and

K(b,a) are 0.5.

slide-61
SLIDE 61

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Density

Odd alphabet

◮ If Σ = {1, b}, the density

  • f the letter b for the

minimal smooth word is given by 1/( √ 2b − 1 − 1)

◮ If Σ = {a, b}, a = 1 : still

an open problem.

Even alphabet

◮ The densities of the letters

a and b are 0.5.

◮ It follows that the densities

  • f the letters in K(a,b) and

K(b,a) are 0.5.

slide-62
SLIDE 62

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Density

Odd alphabet

◮ If Σ = {1, b}, the density

  • f the letter b for the

minimal smooth word is given by 1/( √ 2b − 1 − 1)

◮ If Σ = {a, b}, a = 1 : still

an open problem.

Even alphabet

◮ The densities of the letters

a and b are 0.5.

◮ It follows that the densities

  • f the letters in K(a,b) and

K(b,a) are 0.5.

slide-63
SLIDE 63

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Open problems

◮ Extremal smooth words for different parities alphabet : Lyndon

factorization ? closure properties ? combinatorial properties ? Efficient algorithm ?

◮ Density of letters in m{a,b}, a, b odd, with a = 1 ? ◮ Complexity of extremal smooth words ? ◮ Extremal smooth words for larger k-letter alphabets ?

slide-64
SLIDE 64

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Open problems

◮ Extremal smooth words for different parities alphabet : Lyndon

factorization ? closure properties ? combinatorial properties ? Efficient algorithm ?

◮ Density of letters in m{a,b}, a, b odd, with a = 1 ? ◮ Complexity of extremal smooth words ? ◮ Extremal smooth words for larger k-letter alphabets ?

slide-65
SLIDE 65

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Open problems

◮ Extremal smooth words for different parities alphabet : Lyndon

factorization ? closure properties ? combinatorial properties ? Efficient algorithm ?

◮ Density of letters in m{a,b}, a, b odd, with a = 1 ? ◮ Complexity of extremal smooth words ? ◮ Extremal smooth words for larger k-letter alphabets ?

slide-66
SLIDE 66

Extremal generalized smooth words . . .

  • G. Paquin, S. Brlek,
  • D. Jamet

Outlines Introduction Kolakoski word Run-length encoding Smooth words Generalized Kolakoski words Problems Previous results Over Σ = {1, 2} Over Σ = {1, 3} Extremal words over same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization Density Open problems

Open problems

◮ Extremal smooth words for different parities alphabet : Lyndon

factorization ? closure properties ? combinatorial properties ? Efficient algorithm ?

◮ Density of letters in m{a,b}, a, b odd, with a = 1 ? ◮ Complexity of extremal smooth words ? ◮ Extremal smooth words for larger k-letter alphabets ?