  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.

Combinatorics