More fun with transforms

[Christian G. Bower]

One interpretation of the inverse Moebius (sum of divisors)
transform is that if b = imob a, and a is a sequence of
the number of unlabeled a-structures on n points then b is
the number of unlabeled structures on n points built out
of identical copies of a-structures.

What if I wanted to build out of a-structures that are all
the same size (i.e. the same number of points) but not
necessarily identical. Easy:

b_n = Sum_{i*j=n} a_i*binomial(a_i+j-1,j)

I'll call this transform "samesize"

So the next step is to look for examples in the EIS. I found
a few.

A007503 samesize of (2,2,2,2,...)
A081543 samesize of (1,2,3,4,...)

It appears Benoit Cloitre has been working with this idea.
These two sequences give a g.f. for this transform:

B(x) = Sum_{n>=1} (x^n/(1-x^n)^a_n)

which looks suspisciously like the Euler transform. All we would
have to do is change Sum to Product.

Thus samesize(a_1,a_2,a_3,a_4,...) =
euler(a_1,0,0,0,...) + euler(0,a_2,0,0,...) + euler(0,0,a_3,0,...) + ...

This suggests a metatransform which I'll call slice.

(slice T)(a_0,a_1,a_2,a_3,...) =
T(a_0,0,0,0,...) + T(0,a_1,0,0,...) + ...

and we have samesize = slice euler.

Does anyone remember when we were discussing Witt vectors and the
Somos and Witt transforms were created?

On 8/23/02 I wrote:
---------------
For a sequence a[]

Let b[n] = SUM ( L(n/d, a[d]) ), d|n

where L(n, z) is the number of Lyndon words (aperiodic necklaces) with n beads
of z colors
= 1/n SUM ( mu(n/d)*z^d ), d|n

[Note that z is not necessarily an integer, but n is a nonnegative integer.]

I call b[] the inverse Witt transform of a[] and a[] the Witt transform of
b[].

The Somos transform is now just the Euler transform of the inverse Witt
transform.
---------------

 From this inverse witt = slice CHK

CHK from
http://www.research.att.com/~njas/sequences/transforms2.html

which I'm thinking of resurrecting as the Lyndon transform
because of its connection to Lyndon words.

Other slice examples:

A038041 slice exp (1,1,1,1,...)
A061095 slice AIJ (1,1,1,1,...)

slice is idempotent i.e. slice(slice T)=slice T
thus it is not reversible
slice somos = slice invert
If T is a linear transform i.e. T(a+b)=T(a)+T(b) then slice T=T.

Christian