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SUMMARY:Benoit Cloitre
DTSTART:20260623T130000Z
DTEND:20260623T140000Z
DTSTAMP:20260612T014521Z
UID:CombinatoricsOnWords/140
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Combinatoric
 sOnWords/140/">The Thue-Morse Transform</a>\nby Benoit Cloitre as part of 
 One World Combinatorics on Words Seminar\n\n\nAbstract\nWe define a transf
 orm $T$ on binary words. Given a binary word\, we use the positions of its
  zeros and ones to build a new binary word. Applied to the alternating wor
 d $a_0 = 0101\\ldots$\, the transform gives the Thue-Morse word. We then s
 tudy the orbit $a_m = T^m(a_0)$\, together with the sequences $u_m$ and $v
 _m$ giving the positions of the ones and the zeros in $a_m$. We obtain an 
 explicit formula for $a_m(n)$ in terms of the binary digits of $n$ and $m-
 1$. From this formula we derive Prouhet-Tarry-Escott identities\, composit
 ion formulas that generalize the identities for evil and odious numbers\, 
 and a recurrence formula for the factor complexity of $a_m$. We end with a
  few directions\, such as applying the transform to the Fibonacci word\, w
 hich yields the Fibonacci-Thue-Morse word.\n
LOCATION:https://researchseminars.org/talk/CombinatoricsOnWords/140/
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