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.

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.

The operation which sends a finite set $S$ to the 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.

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.