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Necklace ring

## Summary

In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials.

## Definition

If A is a commutative ring then the necklace ring over A consists of all infinite sequences ${\displaystyle (a_{1},a_{2},...)}$  of elements of A. Addition in the necklace ring is given by pointwise addition of sequences. Multiplication is given by a sort of arithmetic convolution: the product of ${\displaystyle (a_{1},a_{2},...)}$  and ${\displaystyle (b_{1},b_{2},...)}$  has components

${\displaystyle \displaystyle c_{n}=\sum _{[i,j]=n}(i,j)a_{i}b_{j}}$

where ${\displaystyle [i,j]}$  is the least common multiple of ${\displaystyle i}$  and ${\displaystyle j}$ , and ${\displaystyle (i,j)}$  is their greatest common divisor.

This ring structure is isomorphic to the multiplication of formal power series written in "necklace coordinates": that is, identifying an integer sequence ${\displaystyle (a_{1},a_{2},...)}$  with the power series ${\displaystyle \textstyle \prod _{n\geq 0}(1{-}t^{n})^{-a_{n}}}$ .