Witt's formula and montonicity

[wouter meeussen]

Sorry,

I don't entirely get it :
with:
moree[li : {__Integer}] :=
  Module[{n = Plus @@ li},
    Function[nj,
        Fold[#1 + MoebiusMu[#2]Apply[Multinomial, li/#2] &, 0,
          Divisors[nj]]] /@ li]

moree /@ Partitions[4]
produces:

{{0}, {4 - Gamma[7/3]/Gamma[4/3], 4}, {4, 4}, {12 - 8/Pi, 12, 12}, {24, 24, 24, 24}}

So, my interpretation as

Function[nj$, Fold[#1 + MoebiusMu[#2]*Multinomial @@ ({3, 1}/#2) & , 0, Divisors[nj$]]]

is not what was intended.

Are you sure we should sum over all divisors of n_j and (implicitely) sum again over all n_j in
M(n_1,...,n_r)={1\over n}\sum_{d|n_j}\mu(d)(n/d)!/((n_1/d)!...(n_r/d)!)
or should I read
M(n_1,...,n_r)={1\over n}\sum_{d|n  }\mu(d)(n/d)!/((n_1/d)!...(n_r/d)!)
                                  ^^
???
baffled as always,

Wouter.



----- Original Message -----
From: "Olivier Gerard" <ogerard at ext.jussieu.fr>
To: <seqfan at ext.jussieu.fr>
Sent: Saturday, October 25, 2003 1:29 AM
Subject: Fwd: Witt's formula and montonicity


> 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
>
>
>