Equivalence Classes for the μ-Coefficient of Kazhdan–Lusztig Polynomials in Sn

We study equivalence classes relating to the Kazhdan–Lusztig μ(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 μ(x, w) for the permutation groups S 10 and S 11.