The enumeration of permutations sortable by pop stacks in parallel

A stack is a last-in first-out sorting device with push and pop operations. Knuth [7] showed that a permutation π can be sorted (meaning that by applying push and pop operations to the sequence of entries π(1), . . . , π(n) we can output the sequence 1, . . . , n) if and only if π avoids the permutation 231, i.e., if and only if there do not exist three indices 1 ≤ i1 < i2 < i3 ≤ n such that π(i1), π(i2), π(i3) are in the same relative order as 231. Shortly thereafter Tarjan [10], Even and Itai [5], and Pratt [9] studied networks which have multiple stacks either in series or in parallel. The questions typically studied for such networks include: