Fixed Hooks in Arbitrary Columns of Partitions

In a paper by the author, Hemmer, Hopkins, and Keith, the concept of a fixed point in a sequence was applied to the sequence of first column hook lengths of a partition. In this paper, we generalize this notion to fixed hook lengths in an arbitrary column of a partition. We establish combinatorial connections between these fixed hooks and colored partitions that have interesting gap and mex-like conditions. Additionally, we obtain several generating functions for hook lengths of a given fixedness by hook length or part size in unrestricted partitions, as well as some classical restrictions such as odd and distinct partitions.