[seqfan] Re: Lyndon words

[Neil Sloane]

Yes, I always use the alphabet {1,2} when it comes
to listing the words in a binary alphabet. There are
many examples.
Neil


On Sat, Mar 8, 2014 at 8:36 PM, M. F. Hasler <oeis at hasler.fr> wrote:

> Just in case s/o didn't find this one, since it seems not so well Xref'd:
>
> A102659 : List of Lyndon words on {1,2} sorted first by length and
> then lexicographically.
> 1,2,12,112,122,1112,1122,1222,...
>
> This seems to me the most natural encoding of binary Lyndon words.
>
> Of course, no objection on the decimal encoding you proposed
> (yet, an encoding of the above as base-3 numbers seems also quite natural).
>
> Regards,
> Maximilian
>
> On Sat, Mar 8, 2014 at 8:28 PM,  <franktaw at netscape.net> wrote:
> > The empty string is also a Lyndon word, and it can be encoded with the
> first
> > option as "1".
> >
> > "Lyndon words with a leading 1, converted to decimal." works for me.
> >
> > Franklin T. Adams-Watters
> >
> >
> > -----Original Message-----
> > From: Wouter Meeussen <wouter.meeussen at telenet.be>
> > To: seqfan <seqfan at list.seqfan.eu>
> > Sent: Sat, Mar 8, 2014 6:21 pm
> > Subject: [seqfan] Lyndon words
> >
> >
> > In http://oeis.org/A001037  under 'Example' and in
> > http://en.wikipedia.org/wiki/Lyndon_word
> > under 'Enumeration' we find the binary Lyndon words themselves (not their
> > counts) as
> > 0,1,01,001,011,0001,0011,0111,00001,00011,00101,00111,01011,01111,...
> > Since each starts with a zero bit, I can see two ways of encoding these
> as
> > decimal numbers:
> > either prepend a "1" bit
> > {2,3,5,9,11,17,19,23,33,35,37,39,43,47,65,67,69,71,75,77,79,87,95,...}
> > or switch  1's and 0's .
> > {1,0,2,6,4,14,12,8,30,28,26,24,20,16,62,60,58,56,52,50,48,40,32,...}
> > The first seems more natural, and produces an increasing sequence.
> > For such sequence, split in runs by bit-count, the number of Lyndon
> words in
> > each run is A001037.
> >
> > Would you also prefer the first option, and, if so, how should I name it?
> >
> > Wouter.
> >
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-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
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