# Kostka Numbers and Longest Increasing Subsequences

## TL;DR

This paper generalizes a classical bijection linking Kostka numbers, Catalan numbers, and permutations with bounded LIS length, providing new combinatorial interpretations for these numbers.

## Contribution

It introduces a generalized bijection that counts permutations with specific LIS constraints using Kostka numbers, extending previous classical results.

## Key findings

- Kostka numbers count permutations with LIS length at most w
- New bijections relate Kostka numbers to permutations with fixed LIS
- Extended classical combinatorial identities involving Catalan numbers

## Abstract

A classical bijection relates certain Kostka numbers, the Catalan numbers, and permutations of length $n$ with longest increasing subsequence (LIS) of length at most $2.$ We generalize this bijection and find Kostka numbers which count the number of permutations of $n$ with LIS length at most $w,$ the number of permutations with $(1, \cdots, w)$ as a LIS, and other similar subsets of permutations.