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Statistic identifier: St001144

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Collection: Finite Cartan types

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Description: The largest mu-coefficient of the Kazhdan Lusztig polynomial occurring in the Weyl group of given type.

The $\mu$-coefficient of the Kazhdan-Lusztig polynomial $P_{u,w}(q)$ is the coefficient of $q^{\frac{l(w)-l(u)-1}{2}}$ in $P_{u,w}(q)$.

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References: [1]   Warrington, G. S. Equivalence classes for the µ-coefficient of Kazhdan-Lusztig polynomials in $S_n$ [[MathSciNet:2859901]]

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Code:
def statistic(C):
    W = CoxeterGroup(C, implementation='coxeter3')
    r = []
    for u in W:
	U = (W(v) for v in W.bruhat_interval(u, W.long_element()))
        next(U)
        for v in U:
            ldiff = v.length()-u.length()-1
            if is_even(ldiff):
                p = W.kazhdan_lusztig_polynomial(u, v)
                r.append(p[ldiff//2])
    return max(r)


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Statistic values:

['A',1]  => 1
['A',2]  => 1
['B',2]  => 1
['G',2]  => 1
['A',3]  => 1
['B',3]  => 1
['C',3]  => 1
['A',4]  => 1
['B',4]  => 1
['C',4]  => 1
['D',4]  => 1
['F',4]  => 1
['A',5]  => 1
['B',5]  => 2
['C',5]  => 2
['D',5]  => 1
['A',6]  => 1
['A',7]  => 1
['A',8]  => 1
['A',9]  => 5
['A',10] => 28

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Created: Apr 18, 2018 at 22:49 by Martin Rubey

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Last Updated: Apr 18, 2018 at 22:49 by Martin Rubey