The set of forbidden pattern is
 
                          {[1, 2, 3, 4], [1, 3, 2, 4]}
 
                         This is the (RIGOROUS) scheme
 
Warning, new definition for Chi
[{[2, 1], [3, 4, 2, 1], [1, 2, 3], [2, 3, 1, 4], [1, 3, 2], [2, 4, 1, 3],
 
    [3, 4, 1, 2], [3, 1, 2]}, {[], [1], [1, 2], [2, 3, 1]}, table([
 
    [3, 1, 2] = 1
 
    [1, 3, 2] = 2
 
    [2, 4, 1, 3] = 1
 
    [3, 4, 2, 1] = 3
 
    [2, 1] = 1
 
    [3, 4, 1, 2] = 1
 
    [2, 3, 1, 4] = 4
 
    [1, 2, 3] = 3
 
    ]), table([
 
    [2, 3, 1] = {[3, 4, 2, 1], [2, 3, 1, 4], [2, 4, 1, 3], [3, 4, 1, 2]}
 
    [1, 2] = {[1, 2, 3], [1, 3, 2], [2, 3, 1]}
 
    [] = {[1]}
 
    [1] = {[2, 1], [1, 2]}
 
    ]), table([
 
    [3, 1, 2] = {}
 
    [1, 3, 2] = {3}
 
    [2, 3, 1] = {}
 
    [2, 4, 1, 3] = {4}
 
    [1, 2] = {}
 
    [3, 4, 2, 1] = {}
 
    [] = {}
 
    [2, 1] = {}
 
    [3, 4, 1, 2] = {}
 
    [2, 3, 1, 4] = {4}
 
    [1] = {}
 
    [1, 2, 3] = {3}
 
    ])]
 
                              This is the sequence
 
[1, 1, 2, 6, 22, 90, 396, 1837, 8864, 44074, 224352, 1163724, 6129840, 32703074,
 
    176351644, 959658200, 5262988330, 29057961666, 161374413196, 900792925199,
 
    5050924332096, 28434661250454, 160644331001476, 910455895039056,
 
    5174722258676440, 29486753617569684, 168411404890025412, 963888354223500966,
 
    5527261077912111036, 31750323052401948174, 182673854776113608008]
 
                    This is the (Semi-Rigorous) Recurrence
 
[1, 1, 2, 6, 22, 90, 396, 1837, 8864, 44074, 224352, 1163724, 6129840, 32703074,
 
    176351644, 959658200, 5262988330, 29057961666, 161374413196, 900792925199,
 
    5050924332096, 28434661250454, 160644331001476, 910455895039056,
 
    5174722258676440, 29486753617569684, 168411404890025412, 963888354223500966,
 
    5527261077912111036, 31750323052401948174, 182673854776113608008]