@cached_function

def statistic(ct):

    return sum(1 for w in WeylGroup(ct) if is_proper(ct, w))

​

def is_proper(ct, w):

    ct = CartanType(ct)

    n = ct.rank()

    l = w.length()

    d = len(w.descents(side="left"))

    if ct.type() == "A":

        return l <= n + binomial(d + 1, 2)

    if ct.type() in ["B", "C"]:

        return l <= n + d^2

    if ct.type() == "D":

        if d > 3:

            return l <= n + d^2 - d

        return l <= binomial(d + 1, 2)

    if ct.type() == "E":

        m = [0, 1, 3, 6, 12, 20, 36, 63, 120]

        return l <= n + m[d]

    if ct.type() == "F":

        m = [0, 1, 4, 9, 24]

        return l <= n + m[d]

    if ct.type() == "G": # = I_2(6)

        if d == 2:

            return l <= 6 + n

        if d < 2:

            return l <= 2 + d

​

​

def parent_initializer(n):

    cartan_types      = [ ['A',n] ]

    if n >= 2:

        cartan_types += [ ['B',n] ]

    if n >= 3:

        cartan_types += [ ['C',n] ]

    if n >= 4:

        cartan_types += [ ['D',n] ]

    if n in [6,7,8]:

        cartan_types += [ ['E',n] ]

    if n == 4:

        cartan_types += [ ['F',n] ]

    if n == 2:

        cartan_types += [ ['G',n] ]

    return [ CartanType(cartan_type) for cartan_type in cartan_types ]

​

element_repr = repr

levels = [1, 2, 3, 4, 5, 6, 7, 8]

for level in levels:

    for elt in parent_initializer(level):

        print('%s  =>  %s' % (element_repr(elt), statistic(elt)))