# On divisibility by primes in columns of character tables of symmetric groups

## TL;DR

This paper proves that as the size of the symmetric group grows, the proportion of entries divisible by a prime in specific character table columns approaches 100%, using bounds on the number of large $k$-cores.

## Contribution

It establishes a new asymptotic result on divisibility properties of character table entries for symmetric groups, linking prime divisibility to combinatorial core counts.

## Key findings

- Proportion of entries divisible by prime p tends to 1 as n increases.
- Lower bounds on the number of large k-cores are derived.
- Asymptotic behavior of divisibility in character tables is characterized.

## Abstract

For an arbitrary prime $p$ we prove that the proportion of entries divisible by $p$ in certain columns of the character table of the symmetric group $S_n$ tends to 1 as $n\to\infty$. This is done by finding lower bounds on the number of $k$-cores, for $k$ large enough with respect to $n$.