Is there a known expression for, or a nontrivial upper bound on, the number of permutations in $S_k$ with longest increasing subsequence of length at most $n$?

Let $l(\sigma)$ denote the length of the longest increasing subsequence of a permutation $\sigma\in S_k$. It seems like a lot is known about $l(\sigma)$ for a random permutation (and its asymptotic scaling), but are there upper bounds on the number of permutations in $\sigma\in S_k$ with $l(\sigma)\leq n$.

Motivation/context for this question: the moments of traces of random unitaries. It is known that $\int dU |{\rm tr}(U)|^{2k} = k!$ for $k\leq n$, where we integrate over the unitary group $U(n)$ with respect to the Haar measure. More generally, for any $k$ and $n$ one may write the expression as [1] $$ \int dU |{\rm tr}(U)|^{2k} = \sum_{\lambda \vdash k,~\ell(\lambda)\leq n} \chi_\lambda(\mathbb{I})^2\,, $$ summing over integer partitions $\lambda$ of $k$ with length at most $n$, and where $\chi_\lambda(\mathbb{I})$ is the identity character with respect to $\lambda$. The RHS is then counting the number of pairs of Young tableaux with width $\leq n$, which is equivalent to counting the number of permutations in $S_k$ with no increasing subsequences longer than $n$. I'm essentially interested in upper bounds on this quantity which are tighter than the trivial bound of $k!$.

[1] E. Rains, "Increasing Subsequences and the Classical Groups," Electron. J. Comb. 5 (1998) R12. http://eudml.org/doc/119270.