  Given a chain of groups $G_0 \le G_1 \le G_2 ... $, we may form the
corresponding chain of their representation rings, together with induction and
restriction operators. We may let $\textrm{Res}^l$ denote the operator which
restricts down $l$ steps, and similarly for $\textrm{Ind}^l$. Observe then that
$\textrm{Ind}^l \textrm{Res}^l$ is an operator from any particular
representation ring to itself. The central question that this paper addresses
is: "What happens if the $\textrm{Ind}^l \textrm{Res}^l$ operator is a
polynomial in the $\textrm{Ind} \textrm{Res}$ operator?". We show that chains
of wreath products $\{H^n \rtimes S_n\}_{n \in \mathbb{N}}$ have this property,
and in particular, the polynomials that appear in the case of symmetric groups
are the falling factorial polynomials. An application of this fact gives a
remarkable new way to compute characters of wreath products (in particular
symmetric groups) using matrix multiplication. We then consider arbitrary
chains of groups, and find very rigid constraints that such a chain must
satisfy in order for $\textrm{Ind}^l \textrm{Res}^l$ to be a polynomial in
$\textrm{Ind} \textrm{Res}$. Our rigid constraints justify the intuition that
this property is indeed a very rare and special property.
