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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!

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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.

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    $\begingroup$ I did a quick check using the old Chevie package for GAP, and I get the same numbers for those small ranks I actually checked (up to rank $4$ for others than $A$ which I checked up to $6$). I didn't check any more, as I did this very naively, so it would take too long, but this should indicate that there was no error in your code (and the OP has somehow missed a few pairs). $\endgroup$
    – Tobias Kildetoft
    Apr 17, 2018 at 7:14
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    $\begingroup$ @W.Z There are a lot of pairs, so it might be impractical to write them here. But in type $A_3$ there just happens to be precisely 2 pairs with non-zero $\mu$-coefficient and which differ in length by more than $1$, so these are a likely candidate for the missing pairs. Explicitly they are $(s_2,s_2s_1s_3s_2)$ and $(s_1s_3,s_1s_2s_3s_2s_1)$. $\endgroup$
    – Tobias Kildetoft
    Apr 18, 2018 at 6:23
  • $\begingroup$ would you mind adding this to findstat.org/StatisticsDatabase/NewStatistic, ideally with your code? (any package will do, several implementations even better, sage is ideal) $\endgroup$
    – Martin Rubey
    Apr 18, 2018 at 8:51
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    $\begingroup$ This is now findstat.org/St001143 $\endgroup$
    – Martin Rubey
    Apr 19, 2018 at 11:29

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