I am looking for references (both of the readable and of the historical kind!) for the following result (which I formulate in one of its least general forms, so as not to complicate the discussion). I **can** prove all of it, and I have proven much of it on the internet (witt5, witt5f, Exercise 2.9.6 in *Hopf Algebras in Combinatorics*), but this isn't something I want to cite in a paper.

Theorem.Let $\left(b_1,b_2,b_3,...\right)$ be a sequence of integers. Then, the following assertions are equivalent:

Assertion $\mathcal C$:Every positive integer $n$ and every prime divisor $p$ of $n$ satisfy \begin{equation} b_{n\diagup p}\equiv b_{n}\operatorname{mod}p^{v_{p}\left( n\right)} , \end{equation} where $v_p\left(n\right)$ denotes the $p$-adic valuation of $n$ (that is, the greatest integer $m$ such that $p^m$ divides $n$).

Assertion $\mathcal D$:There exists a sequence $\left(x_1,x_2,x_3,...\right)$ of integers such that every positive integer $n$ satisfies $b_n = \sum\limits_{d\mid n}dx_{d}^{n\diagup d}$.

Assertion $\mathcal D^{\prime}$:Same as Assertion $\mathcal D$, but with "a sequence" replaced by "one and only one sequence".

Assertion $\mathcal E$:There exists a sequence $\left(y_1,y_2,y_3,...\right)$ of integers such that every positive integer $n$ satisfies $b_{n}=\sum\limits_{d\mid n}dy_{d}$.

Assertion $\mathcal E^{\prime}$:Same as Assertion $\mathcal E$, but with "a sequence" replaced by "one and only one sequence".

Assertion $\mathcal F$:Every positive integer $n$ satisfies $n\mid \sum\limits_{d\mid n}\mu\left( d\right) b_{n\diagup d}$, where $\mu$ denotes the Möbius function.

Assertion $\mathcal G$:Every positive integer $n$ satisfies $n\mid \sum\limits_{d\mid n}\phi\left( d\right) b_{n\diagup d}$, where $\phi$ denotes Euler's totient function.

Assertion $\mathcal H$:Every positive integer $n$ satisfies $n\mid \sum\limits_{i=1}^{n}b_{\gcd\left( i,n\right) }$.

Assertion $\mathcal I$:There exists a sequence $\left(q_1,q_2,q_3,...\right)$ of integers such that every positive integer $n$ satisfies $b_{n}=\sum\limits_{d\mid n}d\dbinom{q_{d}n\diagup d}{n\diagup d}$.

Assertion $\mathcal I^{\prime}$:Same as Assertion $\mathcal I$, but with "a sequence" replaced by "one and only one sequence".

Assertion $\mathcal J$:There exists a ring homomorphism from the ring $\mathbf{Symm}$ of symmetric functions in infinitely many variables over $\mathbb{Z}$ which sends $p_{n}$ (the $n$-th power sum symmetric function) to $b_{n}$ for every positive integer $n$.

Assertion $\mathcal K$:There exist two sets $M$ and $N$ and two maps $f:M\rightarrow M$ and $g:N\rightarrow N$ such that every positive integer $n$ satisfies \begin{equation} \left\vert \operatorname*{Fix}\left( f^{n}\right) \right\vert <\infty , \qquad \left\vert \operatorname*{Fix}\left( g^{n}\right) \right\vert <\infty \qquad\text{ and }\left\vert \operatorname*{Fix}\left( f^{n}\right) \right\vert -\left\vert \operatorname*{Fix}\left( g^{n}\right) \right\vert =b_{n} , \end{equation} where $\operatorname*{Fix}\left( h\right) $ denotes the set of fixed points of any map $h:S\to S$ (for any set $S$).

I assume more results can be added to this. It should be noticed that each of the assertions $\mathcal D^{\prime}$, $\mathcal E^{\prime}$, $\mathcal I^{\prime}$ follows from the respective un-primed assertion due to $\mathbb Z$ being torsionfree, and that Assertion $\mathcal H$ is more or less a trivial reformulation of Assertion $\mathcal G$.

The above theorem can be seen as a generalization of the famous "necklace divisibility" which states that $n \mid \sum\limits_{d\mid n} \phi\left(d\right)q^{n\diagup d}$ for any positive integer $n$ and any integer $q$. Many similar divisibilites also follow from that theorem. The sequences $\left(b_1,b_2,b_3,...\right)$ which satisfy the equivalent assertions of the Theorem can be called "ghost-Witt vectors over $\mathbb Z$", though the real motivation of this notion comes not from considering sequences of integers but (more generally) families of elements of a commutative ring.

The above theorem is a kind of folklore, except for Assertions $\mathcal I$ and $\mathcal I^{\prime}$ which I have not seen anywhere (but they are sufficiently epigonal that I wouldn't expect them to be new). Lemma 9.93 in Hazewinkel's "Witt vectors. Part 1" (when will there finally be a part 2?) yields $\mathcal C\Longleftrightarrow\mathcal D$. The equivalence of $\mathcal D$, $\mathcal F$, $\mathcal G$ and $\mathcal H$ is stated as the Corollary on page 9 of Andreas Dress, Christian Siebeneicher, *The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, $\lambda$-rings and the universal ring of Witt vectors* (where $\mathcal D$ appears in a generating function form), and a quick look at this paper makes me pretty confidence he proves their equivalence to $\mathcal K$ there. The equivalence between Assertions $\mathcal E$ and $\mathcal F$ follows from Möbius inversion and seems to be implicit in the papers mentioned. I am irked by the fact that I cannot find Assertion $\mathcal J$ explicit in literature, although it is clearly well-known. I have heard that Vladimir Arnold was studying sequences satisfying the equivalent assertions when he was doing discrete dynamical sequences, but I am not sure what a good keyword would be to search for.

Further equivalent assertions that could be added to the list are also welcome!

**UPDATE:** Further references found (thanks to Richard Stanley and Keith Conrad):

The equivalence $\mathcal C \Longleftrightarrow \mathcal D$ is Lemma 1 in Lars Hesselholt,

*Lecture notes on Witt vectors*, 2005. It is credited to Dwork there, although it seems that the forms Dwork stated it in are rather far apart from the form I need.Richard Stanley,

*Enumerative Combinatorics, volume 2*, CUP 2001, exercise 5.2**a**proves the equivalence $\mathcal C \Longleftrightarrow \mathcal F$. Some more of the above assertions appear in equivalent rewritings in that exercise.The closest thing I could find to a proof of $\mathcal D \Longleftrightarrow \mathcal J$ in literature is Christophe Reutenauer,

*On Symmetric Functions Related to Witt Vectors and the Free Lie Algebra*, Advances in Mathematics, vol. 110, issue 2 (February 1995), pp. 234-246.. He never does this equivalence explicitly, but his statement that "the $q_n$ freely generate $\Lambda$ over $\mathbb Z$" (as a commutative algebra) on p. 236, combined with the equality (2.3), yield it very easily.The equivalence $\mathcal C \Longleftrightarrow \mathcal D \Longleftrightarrow \mathcal F$ goes back to Issai Schur,

*Arithmetische Eigenschaften der Potenzsummen einer algebraischen Gleichung*, Compositio Mathematica, 4 (1937), pp. 432-444. There he also proves a statement that can be interpeted as a finitary version of $\mathcal C \Longleftrightarrow \mathcal J$; in fact, he is not considering the ring $\mathbf{Symm}$, but rather he works with actual power sums of the roots of a monic degree-$m$ integer polynomial. Unfortunately it's easier to reprove the equivalence $\mathcal C \Longleftrightarrow \mathcal J$ than derive it from Schur's results, but he clearly would have stated it if he would work with today's notations. Ironically Schur is one of the founding fathers of $\mathbf{Symm}$...Here is a reference from Stanley's EC2 which I could not find: W. Jänischen (I suspect that it should be Jänichen with no "s"),

*Sitz. Berliner Math. Gesellschaft***20**(1921), pp. 23-29. If anyone can send me a scan I'd be very happy. I don't know if "Sitz. Berliner Math. Gesellschaft" is a standalone periodical or a part of "Archiv der Mathematik und Physik" (it used to be the latter at the beginning of the 20th century).Several people refer to A. Dold,

*Fixed point indices of iterated maps*, Inventiones mathematicae, 1983, Volume 74, Issue 3, pp. 419-435. While this is obviously related to assertion $\mathcal K$ (it is mostly about its continuous analoga), I can't see any part of the theorem being proven in this paper. But I've just quickly skimmed the paper.Keith Conrad mentions Donald Knutson,

*$\lambda$-rings and the representation theory of the symmetric group*. Since this text (one of the early good introductions into the representation theory of $S_n$) considers Witt vectors in relation to $\mathbf{Symm}$, it inevitably grazes the above theorem, but it seems to never actually state any part of it. Or have I missed something?

**UPDATE 2:** I just modified Assertion $\mathcal K$. The original version of this assertion was this: "There exists a set $M$ and two maps $f:M\rightarrow M$ and $g:M\rightarrow M$ such that every positive integer $n$ satisfies

$\left\vert \operatorname*{Fix}\left( f^{n}\right) \right\vert <\infty $, $\left\vert \operatorname*{Fix}\left( g^{n}\right) \right\vert <\infty$ and $\left\vert \operatorname*{Fix}\left( f^{n}\right) \right\vert -\left\vert \operatorname*{Fix}\left( g^{n}\right) \right\vert =b_{n}$."

While this original version is equivalent to the new version of Assertion $\mathcal K$, it is a rather unnatural statement (and the equivalence is ugly to prove as far as I can tell).

Enumerative Combinatorics, vol. 2. $\endgroup$