[seqfan] Re: Lyndon words

[franktaw at netscape.net]

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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