# Schur positivity and log-concavity related to longest increasing subsequences

## TL;DR

This paper proves new Schur positivity results related to the log-concavity of generating functions for longest increasing subsequences, offering a novel approach to Chen's conjecture in permutation combinatorics.

## Contribution

It generalizes previous results by establishing Schur positivity, providing a new method to approach Chen's log-concavity conjecture.

## Key findings

- Proved Schur positivity of specific symmetric functions.
- Extended log-concavity results to subsets of the symmetric group.
- Proposed a new approach to Chen's original conjecture.

## Abstract

Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen's log-concavity conjecture, B\'{o}na, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. This also enables us to propose a new approach to Chen's original conjecture.