The problem of stack sorting was introduced by Knuth in 1960s and many variations have been considered since then.
Let $W_t(n,k)$ be the number of $t$-stack sortable $n$-permutations with $k-1$ descents.
Then $W_1(n,k)$ and $W_{n-1}(n,k)$ correspond to the Narayana numbers and the Eulerian numbers, respectively.
In this talk, we show that the numbers $W_t(n,k)$ satisfy central and local limit theorems for $t=1,2,n-1$ and $n-2$.
This result, in particular, gives an affirmative answer to Lou Shapiro's question about the asymptotic normality of the Narayana numbers.
As a generalization, we also show the asymptotic normality of Callan's $m$-th order Narayana numbers.
This talk is based on joint work with Jianxi Mao, Yi Wang, Arthur L.B. Yang, and James J.Y. Zhao.