Fwd: Witt's formula and montonicity

[Olivier Gerard]

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