The purpose of this paper is to present a new non-recursive combinatorial formula for the Kazhdan-Lusztig polynomials of a Coxeter group
W
W
. More precisely, we show that each directed path in the Bruhat graph of
W
W
has a naturally associated set of lattice paths with the property that the Kazhdan-Lusztig polynomial of
u
,
v
u,v
is the sum, over all the lattice paths associated to all the paths going from
u
u
to
v
v
, of
(
−
1
)
Γ
≥
0
+
d
+
(
Γ
)
q
(
l
(
v
)
−
l
(
u
)
+
Γ
(
l
(
Γ
)
)
)
/
2
(-1)^{\Gamma _{\ge 0}+d_+(\Gamma )}q^{(l(v)-l(u)+\Gamma (l(\Gamma )))/2}
where
Γ
≥
0
,
d
+
(
Γ
)
\Gamma _{\ge 0}, d_+(\Gamma )
, and
Γ
(
l
(
Γ
)
)
\Gamma (l(\Gamma ))
are three natural statistics on the lattice path.