Witt vectors and the algebra of necklaces.

*(English)*Zbl 0545.05009If \(A\) is a ring with identity, then a string \(S\) (over \(A\)) is an infinite column finite matrix with associated (unital) series
\[
\prod_{i,j}(1- S_{ij}z^ j)^{-1},\qquad 1\leq i,j<\infty,\quad S=(S_{ij}).
\]
\(S\) and \(S'\) are Witt equivalent if their associated series are identical. Witt equivalence is characterized algorithmically, resulting in Theorem 3: \(S\) and \(S'\) are Witt equivalent if and only if there is a string \(T\) which is obtained from both by “carrying” (the combinatorial process). Sums and products are defined and shown to be compatible with Witt equivalence via arguments involving the associated unital series.

If \(A\) is an integral domain of characteristic 0, the commutative ring obtained is naturally isomorphic to the necklace ring \(\text{Nr}(A)\), the other (combinatorial) object studied in the first part of this very interesting paper. \(\text{Nr}(A)\) consists of infinite vectors \((a_ i)\), with addition defined componentwise and multiplication by an involution \((a_ i)\cdot(b_ j)=(c_ n)\), \(\sum_{[i,j]=n}(i,j)a_ ib_ j\). A study of several natural operators on \(\text{Nr}(A)\) leads to a sequence of useful and elegant results, including a derivation of the formula for the number of primitive necklaces in \(\alpha\) colors and \(n\) beads, \[ M(\alpha;n)=\frac 1n \sum_{d(n)}\mu(n/d)\alpha^ d \] and several other observations, e.g., \(M(\alpha \beta;n)=\sum_{[i,j]=n}M(\alpha;i)M(\beta;j)\).

A necklace is obtained by placing colored beads around a circle. A necklace which is asymmetric under rotation is primitive.

As is noted by the authors, the context developed in this paper allows them to provide a set-theoretic (bijection) proof of the cyclotomic identity \[ (1-\alpha z)^{-1}=\prod_{j\geq 1}(1-z^ j)^{- M(\alpha,j)} \] which is natural in various senses of the word. As always with these authors, this paper is highly readable in addition to being instructive.

If \(A\) is an integral domain of characteristic 0, the commutative ring obtained is naturally isomorphic to the necklace ring \(\text{Nr}(A)\), the other (combinatorial) object studied in the first part of this very interesting paper. \(\text{Nr}(A)\) consists of infinite vectors \((a_ i)\), with addition defined componentwise and multiplication by an involution \((a_ i)\cdot(b_ j)=(c_ n)\), \(\sum_{[i,j]=n}(i,j)a_ ib_ j\). A study of several natural operators on \(\text{Nr}(A)\) leads to a sequence of useful and elegant results, including a derivation of the formula for the number of primitive necklaces in \(\alpha\) colors and \(n\) beads, \[ M(\alpha;n)=\frac 1n \sum_{d(n)}\mu(n/d)\alpha^ d \] and several other observations, e.g., \(M(\alpha \beta;n)=\sum_{[i,j]=n}M(\alpha;i)M(\beta;j)\).

A necklace is obtained by placing colored beads around a circle. A necklace which is asymmetric under rotation is primitive.

As is noted by the authors, the context developed in this paper allows them to provide a set-theoretic (bijection) proof of the cyclotomic identity \[ (1-\alpha z)^{-1}=\prod_{j\geq 1}(1-z^ j)^{- M(\alpha,j)} \] which is natural in various senses of the word. As always with these authors, this paper is highly readable in addition to being instructive.

Reviewer: Joseph Neggers (Tuscaloosa)

##### MSC:

05A10 | Factorials, binomial coefficients, combinatorial functions |

05A15 | Exact enumeration problems, generating functions |

13F35 | Witt vectors and related rings |

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\textit{N. Metropolis} and \textit{G.-C. Rota}, Adv. Math. 50, 95--125 (1983; Zbl 0545.05009)

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