There all together, 2, different equivalence classes

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

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

                                  here it is:

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

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

[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900,

    2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420,

    24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452,

    18367353072152, 69533550916004, 263747951750360, 1002242216651368,

    3814986502092304]

                  This enumerating sequence seems to have the

                                             2 (1 + 2 n)
                   annihilating operator , - ----------- + N
                                                2 + n

For the equivalence class of patterns,

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

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

                                  here it is:

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

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

[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900,

    2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420,

    24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452,

    18367353072152, 69533550916004, 263747951750360, 1002242216651368,

    3814986502092304]

                  This enumerating sequence seems to have the

                                             2 (1 + 2 n)
                   annihilating operator , - ----------- + N
                                                2 + n

                         Out of a total of , 2, cases

                      2, were successful and , 0, failed

                               Success Rate: , 1.

                             Here are the failures

                                       {}

                                       {}

                     It took, 4.044, seconds of CPU time .