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