  Motivated in part by hook-content formulas for certain restricted partitions
in representation theory, we consider the total number of hooks of fixed length
in odd versus distinct partitions. We show that there are more hooks of length
$2$, respectively $3$, in all odd partitions of $n$ than in all distinct
partitions of $n$, and make the analogous conjecture for arbitrary hook length
$t \geq 2$. We also establish additional bias results on the number of gaps of
size $1,$ respectively $2$, in all odd versus distinct partitions of $n$. We
conjecture similar biases and asymptotics, as well as congruences for the
number of hooks of fixed length in odd distinct partitions versus
self-conjugate partitions.
  An integral component of the proof of our bias result for hooks of length $3$
is a linear inequality involving $q(n)$, the number of distinct partitions of
$n$. In this article we also establish effective linear inequalities for $q(n)$
in great generality, a result which is of independent interest.
  Our methods are both analytic and combinatorial, and our results and
conjectures intersect the areas of representation theory, analytic number
theory, partition theory, and $q$-series. In particular, we use a
Rademacher-type exact formula for $q(n),$ Wright's circle method, modularity,
$q$-series transformations, asymptotic methods, and combinatorial arguments.
