  Recently, Blecher and Knopfmacher applied the notion of fixed points to
integer partitions. This has already been generalized and refined in various
ways such as $h$-fixed points for an integer parameter $h$ by Hopkins and
Sellers. Here, we consider the sequence of first column hook lengths in the
Young diagram of a partition and corresponding fixed hooks. We enumerate these,
using both generating function and combinatorial proofs, and find that they
match occurrences of part sizes equal to their multiplicity. We establish
connections to work of Andrews and Merca on truncations of the pentagonal
number theorem and classes of partitions partially characterized by certain
minimal excluded parts (mex).
