1, 1, 3, 11, 49, 263, 1653, 11877, 95991, 862047, 8516221, 91782159, ...
Generating functions
Generating functions
Intertwining two permutations gives a pop-stacked permutation:
$$\Rightarrow p_{2n} \geq (n!)^2$$
$a_{n,k} \dots$ # of 2-pop-stack-sortable permutations of size $n$ with $k$ ascents
$a_{n,k} \dots$ # of 2-pop-stack-sortable permutations of size $n$ with $k$ ascents
Permutation $\pi$ is
$\pi \in \operatorname{Im}(T^{n-2}) \iff {}$ $\pi$ is thin and has no inner runs of odd size.
There are $2^{n/2 - 1} + 2^{n/2} - 1$ and $2^{(n+1)/2} - 1$ such permutations for even and odd $n$, respectively.
Via characterization of $\operatorname{Im}(T^{n-2})$:
skew-layered permutations without odd inner runs