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Identifier
Values
=>
Cc0022;cc-rep
['A',1]=>1 ['A',2]=>4 ['B',2]=>7 ['G',2]=>16 ['A',3]=>10 ['B',3]=>22 ['C',3]=>22 ['A',4]=>20 ['B',4]=>50 ['C',4]=>50 ['D',4]=>28 ['F',4]=>110 ['A',5]=>35 ['B',5]=>95 ['C',5]=>95 ['D',5]=>60 ['A',6]=>56 ['B',6]=>161 ['C',6]=>161 ['D',6]=>110 ['E',6]=>156 ['A',7]=>84 ['B',7]=>252 ['C',7]=>252 ['D',7]=>182 ['E',7]=>399 ['A',8]=>120 ['B',8]=>372 ['C',8]=>372 ['D',8]=>280 ['E',8]=>1240
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Description
The atomic length of the longest element.
The atomic length of an element $w$ of a Weyl group is the sum of the heights of the inversions of $w$.
References
[1] Chapelier-Laget, N., Gerber, T. Atomic length in Weyl groups arXiv:2211.12359
Code
def atomic_length(pi):
    """
    EXAMPLES::

        sage: l = [atomic_length(SignedPermutations(n).long_element()) for n in range(1,8)]
        sage: l
        sage: fricas.guess(l)[0].sage().factor()
        1/6*(4*n + 3)*(n + 2)*(n + 1)

    """
    W = WeylGroup(pi.parent().coxeter_type())
    w = W.from_reduced_word(pi.reduced_word())
    return sum(a.height() for a in w.inversions(inversion_type="roots"))

def statistic(ct):
    return atomic_length(WeylGroup(ct).long_element())

Created
Nov 23, 2022 at 16:34 by Martin Rubey
Updated
Nov 23, 2022 at 16:34 by Martin Rubey