On $t$-core towers and $t$-defects of partitions

TL;DR

This paper investigates the combinatorial structure of t-core towers of partitions, deriving generating functions, asymptotic formulas, and congruences related to the sizes and defects of these partitions.

Contribution

It introduces new generating functions for t-core towers and provides asymptotic and congruence results for the t-defect of partitions, advancing understanding of partition core structures.

Findings

Derived generating functions for sizes of t-core towers

Established asymptotic formulas for average t-defect size

Discovered interesting congruences related to partition cores

Abstract

We study generating functions which count the sizes of $t$-cores of partitions, and, more generally, the sizes of higher rows in $t$-core towers. We then use these results to derive an asymptotic for the average size of the $t$-defect of partitions, as well as some curious congruences.