Lambda-operations on stable homotopy groups of spheres The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable homotopy groups of spheres $\pi_*^s$.
If $A$ is a commutative ring, $K_0(A)$ has a simple definition as the free abelian group of projective finitely generated $A$-modules modulo exact sequences. On this group we use the exterior powers $\Lambda^k$ to get so-called Lambda-operations $\lambda^k$. These have nice properties and one can use them to alternatively construct Adams operations $\Psi^i$. This construction can be extended to all $K_n(A)$, giving $K_*(A)$ the structure of a Lambda-ring. This can found in sections II.4 and IV.5 of Weibel's book.
There is a strong analogy between finite sets and vector spaces. This tells you that an analogue of the exterior power $\Lambda^k$ is given by construction that sends a finite set $X$ to its set of $k$-element subsets ${X \choose k}$. This gives the standard Lambda-ring structure on $\mathbb{Z} = \pi_0^s$, i.e. the one on Wikipedia. 
It seems that Weibel's construction of the Lambda-operations on higher K-theory groups works in this context as well. Is this correct? If so, we get $\lambda^i$ and $\Psi^i$ on the stable homotopy groups of spheres. What is known about these? Have they been used for anything?
 A: You can refine this. Let's take $k=2$ to give the idea. To a based set $X$ you can associate $(X\wedge X)/X$, a based set with free action of $\Sigma_2$. This leads to an operation going from stable homotopy of $S^0$ to stable homotopy of $B\Sigma_2$, such that when followed by transfer it gives the difference between the identity and squaring.
This leads to a proof of the Kahn-Priddy Theorem, a proof due to Kahn and Priddy I believe. (EDIT: No, I guess it was Segal.)
Waldhausen adapted the same idea to prove an important result about his $A(X)$, the "vanishing of the mystery homology theory". (That's why I know about it.) This involved extending the construction of these operations from the algebraic $K$-theory of sets to the algebraic $K$-theory of spaces. 
A: The operation which sends a finite set $S$ to its set of $k$-element subsets, $\binom{S}{k}$, gives rise to the $k$-th stable Hopf invariant. There is additional structure in this: the set $\binom{S}{k}$ has a canonical $k$-fold covering so the operation is better viewed as a map
$$
QS^0 \to Q(B\Sigma_k)_+
$$
rather than as a map 
$$
Q S^0 \to QS^0 ,
$$ 
where for a based space $X$, the space $QX$ is $\Omega^\infty\Sigma^\infty X$
is the representing space for the stable homotopy of $X$, i.e., 
$\pi_j(QX) = \pi_j^{\text{st}}(X)$. So the operation induces a homomorphism
$$
\pi_j^{\text{st}}(S^0)  \to \pi_j^{\text{st}}((B\Sigma_k)_+) .
$$
These operations satisfy certain axioms (Cartan Formula, compatibility with transfers, etc.).
A good place to read about these operations is:
Segal, Graeme: Operations in stable homotopy theory. New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), pp. 105–110. London Math Soc. Lecture Note Ser., No. 11, Cambridge Univ. Press, London, 1974.
A: Dear Sander
Maybe this paper by Pierre Guillot can help you:
http://arxiv.org/abs/math/0612327
