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 (1) Damien Jamet (2) Genevi` Sreˇ Previous results Over Σ = { 1 , 2 } Over Σ = { 1 , 3 } (1) LaCIM - Laboratoire de Combinatoire et d’Informatique Math´ ematique UQAM - Universit´ e du Qu´ ebec ` a Montr´ eal Extremal words over (2) LORIA - INRIA Lorraine same parity alphabets Periodicity Recurrence and closure under reversal Lyndon factorization web : http://www.labmath.uqam.ca/˜ paquin/ Density email : [paquin,brlek]@lacim.uqam.ca, jamet@lirmm.fr Open problems September 2, 2006
Extremal generalized smooth words . . . G. Paquin, S. Brlek, D. Jamet Introduction Kolakoski word Outlines Run-length encoding Introduction Smooth words Kolakoski word Run-length encoding Generalized Kolakoski words Smooth words Problems Generalized Kolakoski words Problems Previous results Previous results Over Σ = { 1 , 2 } Over Σ = { 1 , 2 } Over Σ = { 1 , 3 } Over Σ = { 1 , 3 } Extremal words over same parity alphabets Periodicity Recurrence and Extremal words over same parity alphabets closure under reversal Lyndon factorization Periodicity Density Recurrence and closure under reversal Open problems Lyndon factorization Density Open problems
Extremal generalized Kolakoski word smooth words . . . G. Paquin, S. Brlek, D. Jamet Outlines ◮ Kolakoski word (Kolakoski W., 1965) : Introduction Kolakoski word Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965). Run-length encoding Smooth words K = 22112122122112112212112122112112122122112122121121122 · · · Generalized Kolakoski words = 22112122122112112212112122112112122122112122121121122 · · · Problems Previous results ◮ (Dekking F. M., 1980-1981) : Over Σ = { 1 , 2 } Dekking F.M., On the structure of self generating sequences, S´ em. de th´ eorie des nombres de Bordeaux (1980-1981). Over Σ = { 1 , 3 } ◮ Density of the letters in K ? Extremal words over ◮ Recurrence ? same parity alphabets ◮ Closure under reversal ? Periodicity Recurrence and ◮ (Weakley W. D., 1989) : closure under reversal Lyndon factorization Weakley W. D., On the number of C ∞ -words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989). Density log 3 log 2 ). Open problems The complexity function of K is in O ( n ◮ (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
Extremal generalized Kolakoski word smooth words . . . G. Paquin, S. Brlek, D. Jamet Outlines ◮ Kolakoski word (Kolakoski W., 1965) : Introduction Kolakoski word Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965). Run-length encoding Smooth words K = 22112122122112112212112122112112122122112122121121122 · · · Generalized Kolakoski words = 22112122122112112212112122112112122122112122121121122 · · · Problems Previous results ◮ (Dekking F. M., 1980-1981) : Over Σ = { 1 , 2 } Dekking F.M., On the structure of self generating sequences, S´ em. de th´ eorie des nombres de Bordeaux (1980-1981). Over Σ = { 1 , 3 } ◮ Density of the letters in K ? Extremal words over ◮ Recurrence ? same parity alphabets ◮ Closure under reversal ? Periodicity Recurrence and ◮ (Weakley W. D., 1989) : closure under reversal Lyndon factorization Weakley W. D., On the number of C ∞ -words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989). Density log 3 log 2 ). Open problems The complexity function of K is in O ( n ◮ (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
Extremal generalized Kolakoski word smooth words . . . G. Paquin, S. Brlek, D. Jamet Outlines ◮ Kolakoski word (Kolakoski W., 1965) : Introduction Kolakoski word Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965). Run-length encoding Smooth words K = 22112122122112112212112122112112122122112122121121122 · · · Generalized Kolakoski words = 22112122122112112212112122112112122122112122121121122 · · · Problems Previous results ◮ (Dekking F. M., 1980-1981) : Over Σ = { 1 , 2 } Dekking F.M., On the structure of self generating sequences, S´ em. de th´ eorie des nombres de Bordeaux (1980-1981). Over Σ = { 1 , 3 } ◮ Density of the letters in K ? Extremal words over ◮ Recurrence ? same parity alphabets ◮ Closure under reversal ? Periodicity Recurrence and ◮ (Weakley W. D., 1989) : closure under reversal Lyndon factorization Weakley W. D., On the number of C ∞ -words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989). Density log 3 log 2 ). Open problems The complexity function of K is in O ( n ◮ (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
Extremal generalized Kolakoski word smooth words . . . G. Paquin, S. Brlek, D. Jamet Outlines ◮ Kolakoski word (Kolakoski W., 1965) : Introduction Kolakoski word Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965). Run-length encoding Smooth words K = 22112122122112112212112122112112122122112122121121122 · · · Generalized Kolakoski words = 22112122122112112212112122112112122122112122121121122 · · · Problems Previous results ◮ (Dekking F. M., 1980-1981) : Over Σ = { 1 , 2 } Dekking F.M., On the structure of self generating sequences, S´ em. de th´ eorie des nombres de Bordeaux (1980-1981). Over Σ = { 1 , 3 } ◮ Density of the letters in K ? Extremal words over ◮ Recurrence ? same parity alphabets ◮ Closure under reversal ? Periodicity Recurrence and ◮ (Weakley W. D., 1989) : closure under reversal Lyndon factorization Weakley W. D., On the number of C ∞ -words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989). Density log 3 log 2 ). Open problems The complexity function of K is in O ( n ◮ (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
Extremal generalized Kolakoski word smooth words . . . G. Paquin, S. Brlek, D. Jamet Outlines ◮ Kolakoski word (Kolakoski W., 1965) : Introduction Kolakoski word Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965). Run-length encoding Smooth words K = 22112122122112112212112122112112122122112122121121122 · · · Generalized Kolakoski words = 22112122122112112212112122112112122122112122121121122 · · · Problems Previous results ◮ (Dekking F. M., 1980-1981) : Over Σ = { 1 , 2 } Dekking F.M., On the structure of self generating sequences, S´ em. de th´ eorie des nombres de Bordeaux (1980-1981). Over Σ = { 1 , 3 } ◮ Density of the letters in K ? Extremal words over ◮ Recurrence ? same parity alphabets ◮ Closure under reversal ? Periodicity Recurrence and ◮ (Weakley W. D., 1989) : closure under reversal Lyndon factorization Weakley W. D., On the number of C ∞ -words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989). Density log 3 log 2 ). Open problems The complexity function of K is in O ( n ◮ (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
Extremal generalized Kolakoski word smooth words . . . G. Paquin, S. Brlek, D. Jamet Outlines ◮ Kolakoski word (Kolakoski W., 1965) : Introduction Kolakoski word Kolakoski W., Self Generating Runs, American Math. Monthly 72, Problem 5304 (1965). Run-length encoding Smooth words K = 22112122122112112212112122112112122122112122121121122 · · · Generalized Kolakoski words = 22112122122112112212112122112112122122112122121121122 · · · Problems Previous results ◮ (Dekking F. M., 1980-1981) : Over Σ = { 1 , 2 } Dekking F.M., On the structure of self generating sequences, S´ em. de th´ eorie des nombres de Bordeaux (1980-1981). Over Σ = { 1 , 3 } ◮ Density of the letters in K ? Extremal words over ◮ Recurrence ? same parity alphabets ◮ Closure under reversal ? Periodicity Recurrence and ◮ (Weakley W. D., 1989) : closure under reversal Lyndon factorization Weakley W. D., On the number of C ∞ -words of each length, J. Comb. Theory Series A 51 no.1 , p.55-62 (1989). Density log 3 log 2 ). Open problems The complexity function of K is in O ( n ◮ (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
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