Fixed hooks in arbitrary columns

TL;DR

This paper generalizes the concept of fixed points in partition hook lengths to arbitrary columns, establishing combinatorial links with colored partitions and deriving generating functions for various partition classes.

Contribution

It introduces the notion of fixed hook lengths in any column of a partition and explores their combinatorial properties and generating functions.

Findings

Established connections between fixed hook lengths and colored partitions.

Derived generating functions for fixed hook lengths in unrestricted and restricted partitions.

Extended classical partition results to new fixedness conditions.

Abstract

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.