A type of sequence exponentiation

[Christian G. Bower]

I recently revisited a sequence operation I wrote about on seqfan
back on 6/23/2000.

It occured to me that it is a type of sequence exponentiation.

The operation which I called EOP at the time, I'll represent using
the exponential symbol ^ here.

b^a = EULER(a *d invEULER(b))

Here *d represents Dirichlet convolution or multiplication of the
Dirichlet generation functions.

It satisfies the exponential property b^(g+h) = b^g * b^h
if * is multiplication of the ordinary g.f.'s (ordinary convolution.)

It's analogous to the scalar equation:

y^x = exp(x*log(y))

Furthermore since EULER and WEIGH are both exponential transforms,
we can replace EULER with WEIGH in the equation.

b^a = EULER(a *d invEULER(b)) = WEIGH(a *d invWEIGH(b))

It also makes it easy to see some connections between the two transforms.

WEIGH(A(x)) = EULER(A(x)-A(x^2))
EULER(A(x)) = WEIGH(A(x)+A(x^2)+A(x^4)+A(x^8)+...)

and more generally, if T and U are exponential transforms then:
T(a) = U(L(a)) for some linear transform L. By linear I mean
L(a+b) = L(a)+L(b). Of course there are cases where T and U have
incompatible domains where the equation will not hold.

I also found a sequence that helped me upgrade my combinatorial
interpretation of this operation.

ID Number: A077285
URL:       http://www.research.att.com/projects/OEIS?Anum=A077285
Sequence:  1,1,3,5,10,15,28,41,69,102,160,231,352,498,732,1027,1470,
           2031,2856,3896,5382,7272,9896,13233,17800,23579,31362,41219,
           54288,70791,92456,119698,155097,199512,256664,328134,419436,
           533162,677412,856573
Name:      Number of partitions of n with designated summands.
...
Example:   a(3)=5 because the partitions of 3 with designated summands are
3',
              2'1', 1'11, 11'1, 111'
...
Offset:    0

This sequence is (with each sequence from offset 1)
[1 2 3 4 5...]^[1 1 1 1 1...]

I wrote (back in 2000)
--------------
a represents a sequence of unlabeled structures.
b represents the multiplicities we are allowing.

b(n)=1 if I'm allowed to have n copies of the same a-structure.
b(n)=0 if I'm not allowed.
...
Note also that while I restricted b to being a sequence of 1's and 0's, but
the algorithm will give you a result without that restriction. One
interpretation (that seems to pass the sanity check) is that b(n) is the
number of colors of a cluster that has multiplicity n.
--------------
The colors interpretation works, but I have a better one now.
b represents a sequence of unlabeled structures.

Any set of n copies of the same a-structure must be structured as
a b-structure of order n. If b_n=0, (exactly) n copies of an
a-structure is not allowed.

The operation has the following g.f.

Product_{k>=1} (1+B(x^k))^a_k

We have these special cases which can be verified by the g.f.

EULER(a) = [1 1 1 1 1...]^a = [-1 0 0 0 0...]^(-a)
WEIGH(a) = [1 0 0 0 0...]^a
EULER(CYCLE(a)) = (-a)^[-1 -1 -1 -1 -1...]
(i.e. unlabeled permutations of a-structures)

Here CYCLE is the transform called CIK on the page
http://www.research.att.com/~njas/sequences/transforms2.html

Christian