## Enumerative properties of Rooted Weighted Necklaces

**Joaquin Carbonara**

Buffalo State University

[This is joint work with Alwin Green.]

Let C_k be the cyclic group {1,q,...,q^{k-1}} and Z[C_k] the algebra
of polynomials in the variables 1, q, q^2, ..., q^{k-1} with coefficients
in Z. We introduce Weighted Rooted Necklaces, which give a combinatorial
interpretation of Z[C_k]. Then, an invertible, non-linear operator R:N[C_k]
--> N[C_k] ("N" is the set of non-negative integers) is also
introduced: R(f[q]) = q^{-f[0]} ( f[q]-f[0]+\sum_{i=1}^{f[0]}q^i ). The
operator R creates equivalence classes on N[C_k]. We determine a recursive
function to count the equivalence class containing G[q] = \sum_{i=0}^k
q^i in Z[C_k]. We give a closed form for such recursive formula in some
instances. We also develop a series of related questions. The Algebraic
Combinatorics setting we establish here was motivated in the first place
by a "hands on" combinatorics problem first posed by Cipra in
1992 (such problem is still open in full generality.)