There all together, 7, different equivalence classes

    For the equivalence class of patterns, {{[1, 2, 3, 4]}, {[4, 3, 2, 1]}}

             the member , {[1, 2, 3, 4]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

                 Using  the scheme, the first, , 31, terms are

[1, 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705,

    167078577, 1148208090, 8026793118, 56963722223, 409687815151, 2981863943718,

    21937062144834, 162958355218089, 1221225517285209, 9225729232653663,

    70209849031116183, 537935616492552297, 4147342550996290153,

    32159907636432567578, 250717538500344886206, 1964347085978431234383,

    15462159345628498316319, 122238900487877503161969]

                  This enumerating sequence seems to have the

                                         2                    2
                                9 (n + 1)    (41 + 42 n + 10 n ) N    2
        annihilating operator , ---------- - --------------------- + N
                                        2                 2
                                 (n + 4)           (n + 4)

For the equivalence class of patterns,

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

             the member , {[1, 2, 4, 3]}, has a scheme of depth , 4

                                  here it is:

{[[], {}, {}], [[1], {}, {}], [[2, 1], {}, {1}], [[1, 3, 2], {}, {2}],

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

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

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

                 Using  the scheme, the first, , 31, terms are

[1, 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705,

    167078577, 1148208090, 8026793118, 56963722223, 409687815151, 2981863943718,

    21937062144834, 162958355218089, 1221225517285209, 9225729232653663,

    70209849031116183, 537935616492552297, 4147342550996290153,

    32159907636432567578, 250717538500344886206, 1964347085978431234383,

    15462159345628498316319, 122238900487877503161969]

                  This enumerating sequence seems to have the

                                         2                    2
                                9 (n + 1)    (41 + 42 n + 10 n ) N    2
        annihilating operator , ---------- - --------------------- + N
                                        2                 2
                                 (n + 4)           (n + 4)

                         Out of a total of , 7, cases

                      2, were successful and , 5, failed

                             Success Rate: , 0.286

                             Here are the failures

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

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

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

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

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

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

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

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

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

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

                    It took, 9387.794, seconds of CPU time .