The set of forbidden pattern is
 
                          {[1, 2, 3, 4], [1, 2, 4, 3]}
 
                         This is the (RIGOROUS) scheme
 
Warning, new definition for Chi
[{[2, 1], [1, 2, 3], [1, 3, 2], [3, 1, 2], [2, 4, 1, 3], [3, 4, 1, 2],
 
    [3, 4, 2, 1], [2, 3, 1, 4]}, {[], [1], [1, 2], [2, 3, 1]}, table([
 
    [2, 3, 1, 4] = 4
 
    [3, 4, 1, 2] = 1
 
    [3, 1, 2] = 1
 
    [3, 4, 2, 1] = 3
 
    [1, 3, 2] = 2
 
    [2, 1] = 1
 
    [2, 4, 1, 3] = 1
 
    [1, 2, 3] = 3
 
    ]), table([
 
    [1, 2] = {[1, 2, 3], [2, 3, 1], [1, 3, 2]}
 
    [] = {[1]}
 
    [2, 3, 1] = {[2, 4, 1, 3], [3, 4, 1, 2], [3, 4, 2, 1], [2, 3, 1, 4]}
 
    [1] = {[2, 1], [1, 2]}
 
    ]), table([
 
    [2, 3, 1, 4] = {3, 4}
 
    [3, 4, 1, 2] = {}
 
    [1, 2] = {}
 
    [3, 1, 2] = {}
 
    [3, 4, 2, 1] = {}
 
    [1, 3, 2] = {}
 
    [] = {}
 
    [2, 3, 1] = {}
 
    [2, 1] = {}
 
    [2, 4, 1, 3] = {}
 
    [1] = {}
 
    [1, 2, 3] = {2, 3}
 
    ])]
 
                              This is the sequence
 
[1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,
 
    142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926,
 
    3236724317174, 17518619320890, 95149655201962, 518431875418926,
 
    2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366,
 
    2588758890960637798, 14308406109097843626, 79228031819993134650]
 
                    This is the (Semi-Rigorous) Recurrence
 
                                                2      2
                        -2 + n + 3 N - 6 n N + N  + n N