**Identifier**

Identifier

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

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 $P_{u,w}(q)$.

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 $P_{u,w}(q)$.

References

[1]

[2]

**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:2859901Code

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

Created

Apr 18, 2018 at 22:32 by

**Martin Rubey**Updated

Apr 18, 2018 at 22:32 by

**Martin Rubey**searching the database

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