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Statistic identifier: St001143

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Collection: Finite Cartan types

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Description: The number of pairs in the Weyl group of given type with mu-coefficient of the Kazhdan Lusztig polynomial being non-zero.

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]   Vogan, D. Number of pairs of permutation in $S_n$ whose µ-coefficient (of their Kazhdan Lusztig polynomial) is non-zero [[MathOverflow:298028]]
[2]   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):
    """                                                                                                                                               
    sage: statistic(CartanType(["A", 4]))                                                                                                             
    482                                                                                                                                               
    """
    W = CoxeterGroup(C, implementation='coxeter3')
    r = 0
    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)
		if p[ldiff//2] != 0:
                    r += 1
    return r


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Statistic values:

['A',1] => 1
['A',2] => 8
['B',2] => 12
['G',2] => 20
['A',3] => 60
['B',3] => 152
['C',3] => 152
['A',4] => 482
['B',4] => 2148
['C',4] => 2148
['D',4] => 892
['F',4] => 8920
['A',5] => 4268
['B',5] => 35070
['C',5] => 35070
['D',5] => 14874
['A',6] => 41934
['B',6] => 679152
['C',6] => 679152
['D',6] => 287438
['E',6] => 846476
['A',7] => 457782

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Created: Apr 18, 2018 at 22:32 by Martin Rubey

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Last Updated: Apr 18, 2018 at 22:32 by Martin Rubey