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).

  • $\begingroup$ Some partial information (with references) can be found in Exercise 5.2 of Enumerative Combinatorics, vol. 2. $\endgroup$ Apr 10, 2013 at 0:21
  • $\begingroup$ Thanks! EC2 is actually in my list of references, but not for Exercise 5.2 which I failed to notice. It handles $\mathcal C \Longleftrightarrow \mathcal F$ at least, and it gives interesting references. $\endgroup$ Apr 10, 2013 at 1:52

1 Answer 1


This question is a bit hard to answer for a few reasons. As you say, most of the equivalences are implicit in the literature. So then it becomes a question of how explicit you want them to be stated, and that's a matter of personal taste. Also, some of them were known earlier in the p-typical case (going all the way back to Witt work in the 30s), although the same ideas will work in the "big" case. So whether such things count is also not clear.

That said, here are my reactions, although you probably know much of what I have to say. I believe all the references I mention are given in my paper "Basic geometry of Witt vectors, I".

For D=J, this is of course a consequence of the fact that big Witt vectors and lambda-rings are two sides to the same coin. So it will be at least implicit in any reference that mentions both of them, such as the chapter in Bourbaki, Hazewinkel's book, or Joyal's two papers in the Canadian Comptes Rendus. There is also Witt's unpublished note from 60s which appears in his collected works. From what I remember, he doesn't talk about lambda-rings, but he talks about Witt vectors and power series, so it's surely at least implicit there too. (I'm also interested in the question of who first realized that Witt vectors and lambda-rings are the same thing. I've spoken to Cartier and Serre about it, and neither is sure. This does appear in the Grothendieck-Mumford correspondence in Mumford's selected papers II page 692. See the letter to Mumford 31 August 1964. Perhaps Grothendieck was the first to realize that the two might be related.)

For D=D', this is of course a consequence of the fact that the ghost map is injective for torsion-free rings. So again, it will be implicit in just about any reference that mentions that fact. Aside from the references above, there is Bergman's chapter in Mumford's book. There are some exercises in the first edition of Lang's Algebra (1969?) which discuss big Witt vectors. Perhaps it's implicit there as well.

For C/E/E'=D/D'/J, I think of this as a special instance of the linear description of Witt vectors of lambda-rings. I believe this is in Joyal. (Update: It's only there in the p-typical case.) I'm not sure if it appeared anywhere earlier.

The equivalence of K and the others is, as you say, presumably just a disguised form of the description of W(Z) as the Burnside ring and hence will at least be implicit in Dress-Siebeneicher. That's probably the first appearance of W(Z)=Burnside; so I think it's unlikely to be anywhere earlier.

  • $\begingroup$ Wow, that's a ton of references which are new to me, including Joyal's (which I must have missed due to their absence from Hazewinkel's list of references and from the internet). Thanks a lot! I don't really care about equivalences like D=D', E=E' etc. which, as you said, are more or less trivial; the reason why I was listing them as separate assertions is to later point out the differences with non-torsionfree cases in what I'm writing. When you refer to "the first edition of Lang's Algebra", do you explicitly mean "not the later editions", as in, the problems got retracted? $\endgroup$ Apr 15, 2013 at 17:50
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    $\begingroup$ Also, here is the text from Grothendieck's letter to Mumford (thanks to a friend): "I liked also Bergman's Chap 26-27, and especially his universal Witt scheme, realized as a formal power series functor. This meets with some old ponderings of mine on power series beginning with 1, on which I make some comments in my little paper on Chern classes (appendix to Borel-Serre). As i point out there this is not only a ring, but a \lambda ring (and even a special \lambda-ring),... $\endgroup$
    – JBorger
    Apr 16, 2013 at 3:04
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    $\begingroup$ on the other hand since Gabriel's seminar on formal groups I had the feeling that the Witt rings must also have a \lambda structure (or something very close to it). Namely, according to Dieudonne-Cartier-Gabriel, certain algebras over W_{infty}(k) (the Witt vector ring over a perfect field k) allow to classify, either commutative formal groups without toroidal part (one might call them ind-unipotent), or ordinary unipotent algebraic groups (the two classifications being in fact dual), in terms of modules over these algebras. Now, in the categories in question, ... $\endgroup$
    – JBorger
    Apr 16, 2013 at 3:04
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    $\begingroup$ one not only has a structure of abelian category, but also the notion of tensor product and consequently of exterior power. Now this extra structure should be reflected in some extra structure of the mentioned classifying algebra, and presumably, en derniere analyse, by W itself. This question should certainly be investigated some day,and perhaps Bergman has a good starting point. I guess, besides, you noticed that, analogously, the classifying space of the infinite unitary group of K-theory is not only a group in the hot-category, $\endgroup$
    – JBorger
    Apr 16, 2013 at 3:05
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    $\begingroup$ but actually a \lambda-ring, and the same remark applies to the orthogonal case, these facts reflecting simply the \lambda-ring structure of the K-functors." $\endgroup$
    – JBorger
    Apr 16, 2013 at 3:05

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