#The set of 4-tuples below is an enumeration scheme for 
#permutations that are 2-stack-sortable in a depth 3 stack.
#
#Each 4-tuple is of the form
#[prefix, gap vectors, reversibly deletable elements, stop points].
#
#The enumeration scheme was discovered after 2 CPU hours
#of work by the bVATTER algorithm available at
#http://faculty.valpo.edu/lpudwell/maple/bVATTER14
#More details about notation and implementation are available in 
#L. Pudwell, "Enumeration Schemes for Permutations Avoiding Barred Patterns", Electronic Journal of Combinatorics 17 (1) (2010), R29. 
#at http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r29
#
#More details about permutations sortable in finite
#depth stacks are available in the paper
#"Sorting Permutations with Finite Depth Stacks"
#by T. Goodrich, D. Groth, L. Knop, W. Olson, L. Pudwell, and J. Yuan
#available at http://faculty.valpo.edu/lpudwell/papers/finite_stack_sorting.pdf

{[[], {}, {}, {}], [[1], {}, {}, {}], [[1, 2], {}, {}, {}], [[2, 1], {}, {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}, {}], [[1, 3, 2], {}, {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {2}, {}], [[2, 3, 1], {}, {}, {}], [[3, 1, 2], {}, {}, {}], [[3, 2, 1], {}, {}, {}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {1}, {}], [[1, 4, 2, 3], {[1, 0, 0, 0, 0]}, {1}, {}], [[1, 4, 3, 2], {[1, 0, 0, 0, 0]}, {1}, {}], [[2, 3, 1, 4], {[0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {1}, {}], [[2, 4, 1, 3], {}, {}, {}], [[2, 4, 3, 1], {}, {}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {2}, {}], [[3, 2, 1, 4], {[0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {2}, {}], [[3, 4, 1, 2], {}, {}, {}], [[3, 4, 2, 1], {}, {}, {}], [[4, 1, 2, 3], {[1, 0, 0, 0, 0]}, {2}, {}], [[4, 1, 3, 2], {[1, 0, 0, 0, 0]}, {2}, {}], [[4, 2, 1, 3], {[1, 0, 0, 0, 0]}, {3}, {}], [[4, 2, 3, 1], {}, {}, {}], [[4, 3, 1, 2], {[1, 0, 0, 0, 0]}, {3}, {}], [[4, 3, 2, 1], {}, {}, {}], [[2, 4, 1, 3, 5], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 4, 3, 1, 5], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 5, 1, 3, 4], {[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 5, 1, 4, 3], {[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 5, 3, 1, 4], {[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}, {}], [[2, 5, 4, 1, 3], {[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}, {}], [[3, 4, 1, 2, 5], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}, {}], [[3, 4, 2, 1, 5], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}, {}], [[3, 5, 1, 2, 4], {[1, 0, 0, 0, 0, 0]}, {3}, {}], [[3, 5, 1, 4, 2], {[1, 0, 0, 0, 0, 0]}, {3}, {}], [[3, 5, 2, 1, 4], {[1, 0, 0, 0, 0, 0]}, {4}, {}], [[3, 5, 2, 4, 1], {[1, 0, 0, 0, 0, 0]}, {5}, {}], [[3, 5, 4, 1, 2], {[1, 0, 0, 0, 0, 0]}, {4}, {}], [[3, 5, 4, 2, 1], {[1, 0, 0, 0, 0, 0]}, {5}, {}], [[4, 2, 3, 1, 5], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {2}, {}], [[4, 3, 2, 1, 5], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {2}, {}], [[4, 5, 1, 2, 3], {[1, 0, 0, 0, 0, 0]}, {3}, {}], [[4, 5, 1, 3, 2], {[1, 0, 0, 0, 0, 0]}, {3}, {}], [[4, 5, 2, 1, 3], {[1, 0, 0, 0, 0, 0]}, {4}, {}], [[4, 5, 2, 3, 1], {[1, 0, 0, 0, 0, 0]}, {5}, {}], [[4, 5, 3, 1, 2], {[1, 0, 0, 0, 0, 0]}, {4}, {}], [[4, 5, 3, 2, 1], {[1, 0, 0, 0, 0, 0]}, {5}, {}], [[5, 2, 3, 1, 4], {[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {2}, {}], [[5, 2, 4, 1, 3], {[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {2}, {}], [[5, 3, 2, 1, 4], {[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {3}, {}], [[5, 3, 4, 1, 2], {[1, 0, 0, 0, 0, 0]}, {4}, {}], [[5, 3, 4, 2, 1], {[1, 0, 0, 0, 0, 0]}, {5}, {}], [[5, 4, 2, 1, 3], {[0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {3}, {}], [[5, 4, 3, 1, 2], {[1, 0, 0, 0, 0, 0]}, {4}, {}], [[5, 4, 3, 2, 1], {[1, 0, 0, 0, 0, 0]}, {5}, {}]}