Fwd: Witt's formula and montonicity

[Pieter Moree]

Dear list,

I promised to get back to my earlier question regarding
the monotonicity of dimensions of free Lie algebras (see below).

This could be dealt with by using Lyndon words and I am
very grateful to Frank Ruskey for suggesting to use these.

A preprint on this is now available at
http://staff.science.uva.nl/~moree/preprints.html
Title: The formal series Witt transform.

Since this paper has little to do with number theory, my usual
area, I had to rely more on the insights and knowledge of other
mathematicians, which many kindly shared.

Thanks !
Pieter Moree


Subject: Fwd: Witt's formula and monotonicity
From: Olivier Gerard <ogerard at ext.jussieu.fr>
Date: Sat, October 25, 2003 12:29 am
To: seqfan at ext.jussieu.fr

Message from Pieter Moree

Dear list,

Let n_1,...,n_r be non-negative numbers.
In Witt's classical dimensional formula for free Lie algebras
the numbers M(n_1,...,n_r) turn up (as dimensions), where

M(n_1,...,n_r)={1\over n}\sum_{d|n_j}\mu(d)(n/d)!/((n_1/d)!...(n_r/d)!)

and n=n_1+...+n_r.

As usual \mu denotes the Moebius function.

I am interested in monotonicity results for these numbers.

The sequence {M(n_1,...,n_r,m)}_{m=0}^{\infty} is a non-decreasing
sequence, provided that n_1+...+n_r > 0.

This was proved by Prof. Bryant (in an e-mail to me)
using the Hall basis for free lie algebras.

I asked already many mathematicians, but none knew of a reference. This
result might well be known.

The number M(n_1,...,n_r) is also the number of circular words of length
n and with primitive period n (aperiodic words of length n) such that
the symbol 1 occurs n_1 times, etc.. (We consider r letters 1,...,r as
an alphabet). Can one also prove the above
result in this setting ?

Define
V(n_1,...,n_r)={(-1)^n\over
n}\sum_{d|n_j}(-1)^{n/d}\mu(d)(n/d)!/((n_1/d)!...(n_r/d)!) (with n as
before)

Is {V(n_1,...,n_r,m)}_{m=0}^{\infty} a non-decreasing sequence
if n_1+...+n_r>0 ?

The latter numbers occur as dimensions of certain free lie superalgebras
(work of Petrogradsky).

Using e.g. Maple it is easy to set up some conjectures regarding
strict monotonicity of the sequences above (the conditions on
n_1,...,n_r need only a little sharpening it seems).

Any comments, references etc. on this issue will be appreciated !

Bests,
Pieter Moree