Warning, the protected name Chi has been redefined and unprotected
             There all together, 8, 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], {}, {}], [[2, 1], {}, {1}], [[1, 1], {}, {1, 2}],

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

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

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

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

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

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

 Using  the scheme, the first, , 10, terms for , 1,  copies of each letter are

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

 Using  the scheme, the first, , 10, terms for , 2,  copies of each letter are

[1, 6, 90, 1879, 47024, 1331664, 41250519, 1367533365, 47808569835,

    1744233181074]

 Using  the scheme, the first, , 7, terms for , 3,  copies of each letter are

           [1, 20, 1680, 173891, 21347262, 2977892253, 455912368540]

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, 1], {}, {1, 2}],

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

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

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

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

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

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

 Using  the scheme, the first, , 10, terms for , 1,  copies of each letter are

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

 Using  the scheme, the first, , 10, terms for , 2,  copies of each letter are

[1, 6, 90, 1879, 47024, 1331664, 41250519, 1367533365, 47808569835,

    1744233181074]

 Using  the scheme, the first, , 7, terms for , 3,  copies of each letter are

           [1, 20, 1680, 173891, 21347262, 2977892253, 455912368540]

                         Out of a total of , 8, cases

                      2, were successful and , 6, failed

                             Success Rate: , 0.250

                             Here are the failures

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

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

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

    {{[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]}}}