Counterexamples to the 0-1 Conjecture

For permutations $x$ and $w$, let $\mu(x,w)$ be the coefficient of highest possible degree in the Kazhdan-Lusztig polynomial $P_{x,w}$. It is well-known that the $\mu(x,w)$ arise as the edge labels of certain graphs encoding the representations of $S_n$. The 0-1 Conjecture states that the $\mu(x,w) \in \{0,1\}$. We present two counterexamples to this conjecture, the first in $S_{16}$, for which $x$ and $w$ are in the same left cell, and the second in $S_{10}$. The proof of the counterexample in $S_{16}$ relies on computer calculations.