I am interested in how many pairs of permutations $(u,w)$ in $S\_n$, such that the $\mu$-coefficient of its Kazhdan-Lusztig polynomial $P\_{u,w}(q)$ is non-zero? where $\mu\_{u,w}=[q^{\frac{l(w)-l(u)-1}{2}}]P\_{u,w}(q)$.


Let's call this number $M\_n$.


I've computed for small $n$'s , but my results seem to be wrong:


$M\_3=8$ $M\_4=58$
$M\_5=480$
$M\_6=4238$


Are there any known properties about this?


Are there any known sufficient conditions for the $\mu$ coefficient to be non-zero (seems to be not)?


Thanks!



Number of pairs of permutation in $S_n$ whose $\mu$-coefficient (of their Kazhdan Lusztig polynomial) is non-zero
When I calculated these numbers using Marc van Leeuwen and Fokko du Cloux's software atlas, I got


* S\_3: 8
* S\_4: 60
* S\_5: 482
* S\_6: 4268
* S\_7: 41934
* S\_8: 457782


(I could easily have made some silly mistake in coding, but I checked the answers by hand in rank 2, and the fact that they're close to yours makes me believe these.) I did the calculations also for some other Weyl groups:


* BC\_2: 12
* BC\_3: 152
* BC\_4: 2148
* BC\_5: 35070
* BC\_6: 679152
* D\_4: 892
* D\_5: 14874
* D\_6: 287438
* G\_2: 20
* F\_4: 8920
* E\_6: 846476


I don't know any simple answers to your questions. The most interesting results about mu are in the work of Greg Warrington and his collaborators; see for example 


Warrington, Gregory S. Equivalence classes for the μ-coefficient of Kazhdan-Lusztig polynomials in Sn. Exp. Math. 20 (2011), no. 4, 457–466. 


