We present a connection between the Leinert sets and the non-crossing two-partitions and we use this connection to give a simple proof of the free Khintchine inequality in discrete non-commutative
L
p
L_p
-spaces. Moreover we extend the inequality of Haagerup-Pisier,
\[
‖
∑
g
∈
S
λ
(
g
)
⊗
a
g
‖
C
λ
∗
(
F
n
)
⊗
A
≤
2
max
{
‖
∑
g
∈
S
a
g
∗
a
g
‖
1
2
,
‖
∑
g
∈
S
a
g
a
g
∗
‖
1
2
}
,
\left \| \sum _{g\in S} \lambda (g)\otimes a_g\right \|_{C_\lambda ^*(F_n)\otimes A} \le 2\max \left \{\left \| \sum _{g\in S} a_g^*a_g\right \|^{\frac 12}, \left \|\sum _{g\in S} a_g a_g^*\right \|^{\frac 12}\right \},
\]
where
λ
\lambda
is the left regular representation of the group
F
n
F_n
,
a
g
a_g
are elements of the
C
∗
C^*
-algebra
A
A
, and
S
S
is the set of the words with length one, to the set
S
S
of the words with arbitrary fixed length.