  Conrey, Farmer, Keating, Rubinstein and Snaith have given a recipe that
conjecturally produces, among others, the full moment polynomial for the
Riemann zeta function. The leading term of this polynomial is given as a
product of a factor explained by arithmetic and a factor explained by
combinatorics (or, alternatively, random matrices). We explain how the lower
order terms arise, and clarify the dependency of each factor on the exponent
$k$ that is considered.
  We use extensively the theory of symmetric functions and representations of
symmetric groups, ideas of Lascoux on manipulations of alphabets, and a key
lemma, due in a basic version to Bump and Gamburd. Our main result ends up
involving dimensions of skew partitions, as studied by Olshanski, Regev,
Vershik, Ivanov and others.
  In this article, we also lay the groundwork for later unification of the
combinatorial computations for lower order terms in the moments conjectures
across families of $L$-functions of unitary, orthogonal and symplectic types.
