Witt's formula and montonicity

[Pieter Moree]

Dear list,

The problems I posed you seem solved (I've the proofs in mind, though
not yet on paper). The magical word is `Lyndon
word' and this was whispered kindly into my ear by Frank Ruskey.
Greatly appreciated, Frank !

The idea Frank suggested is roughly to extend Lyndon words by a letter
of highest lexicographical order, this will yield a Lyndon
word again (e.g. by Proposition 5.1.3 of  M. Lothaire, Combinatorics on
Words, Enc. of Math.). This then gives an injection of the
relevant sets.

Refinements of this argument can be used to yield
strict monotonicity and also to deal with the case with the minus
signs coming in. Some case analysis needs to be done.

I am still looking for references if any for this, apparently pretty
easy result.
If things are written down (as part of
a bigger project) I plan to report back to you.

Regards,
Pieter Moree