A 2-stack pushall sortable permutation is one that can be sorted by two stacks in series by pushing all elements to the stacks before writing any element to the output.

Permutations sortable with two stacks in series A263929.

Pushall stack words of length 3n A274969. A permutation of length n is 2-stack pushall sortable if and only if it can be sorted by a sequence of 3n operations represented by a pushall stack word of length 3n.