Permutations sortable by deques and by two stacks in parallel

In his seminal work The Art of Computer Programming [2], Knuth was the first to consider a number of classic data structures from the point of view of the permutations they could produce from the identity permutation, or equinumerously, the permutations which the data structure can sort. Famously, he noticed that the permutations obtainable using a single stack are exactly those which avoid the pattern 312. Knuth posed the same question for a number of other data structures, in particular, he asked how many permutations of each length can be obtained using a double ended queue (deque) or two stacks in parallel (tsip).