Suppose we have a combinatorial bracelet composed of natural numbers.

What is the number of different bracelets whose elements sum up to a previously fixed natural number N?

Also, are there any results if we add a constraint that the number of beads on the bracelet is always odd?

P.S. Any good upper bounds are also helpful.

(EDITED in the light of the comments below)

Suppose we have a combinatorial bracelet composed of natural numbers.

(Two bracelets are equivalent if you can get from one to the other via rotation or reflection.)

What is the number of different bracelets whose elements sum up to a previously fixed natural number N?

Also, are there any results if we add a constraint that the number of beads on the bracelet is always odd?

P.S. Any good upper bounds are also helpful.

(EDITED in the light of the comments below)

Suppose we have a combinatorial bracelet composed of integers.

What is the number of different bracelets whose elements sum up to a previously fixed integer N?

Also, are there any results if we add a constraint that the number of beads on the bracelet is always odd?

P.S. Any good upper bounds are also helpful.

Suppose we have a combinatorial bracelet composed of natural numbers.

What is the number of different bracelets whose elements sum up to a previously fixed natural number N?

Also, are there any results if we add a constraint that the number of beads on the bracelet is always odd?

P.S. Any good upper bounds are also helpful.

(EDITED in the light of the comments below)