We study equivalence classes relating to the Kazhdan–Lusztig $\mu(x,w)$ coefficients in order to help explain the scarcity of distinct values. Each class is conjectured to contain a “crosshatch” pair. We also compute the values attained by $\mu(x,w)$ for the permutation groups $S_10$ and $S_11$.
"Equivalence Classes for the μ-Coefficient of Kazhdan–Lusztig Polynomials in $S_n$." Experiment. Math. 20 (4) 457 - 466, 2011.