corrigés

n
n
X
k=0 n
k2
=2n
n
(x+y)2n
(x+y)2n
(x+y)2n=
2n
X
k=0 2n
k!x2n−kyk
(x+y)2n=`(x+y)n´2
= n
X
k=0 n
k!xn−kyk!2
(x+y)n
= n
X
i=0 n
i!xn−iyi! n
X
j=0 n
j!xn−jyj!
=
n
X
i=0
n
X
j=0 n
i! n j!xn−i−jyi+j
=X
0≤i≤n
0≤j≤n
n
i! n j!xn−i−jyi+j
=
2n
X
k=0 X i+j=k
0≤i≤n
0≤j≤n
n
i! n j!xn−i−jyi+j{(i, j) 0 ≤i, j ≤n}=
n
[
k=0
{(i, j)i+j=k0≤i, j ≤n}
=
2n
X
k=0 X i+j=k
0≤i≤n
0≤j≤n
n
i! n j!xn−kyk
=
2n
X
k=0
0
B
B
B
B
B
@X
i+j=k
0≤i≤n
0≤j≤n
n
i! n j!1
C
C
C
C
C
A
xn−kykxn−kyk
=
2n
X
k=0
0
B
B
@X
0≤i≤n
0≤k−i≤n
n
i! n k−i!1
C
C
A
xn−kykj7→ k−i
x y
2n
X
k=0 2n
k!x2n−kyk=
2n
X
k=0
0
B
B
@X
0≤i≤n
0≤k−i≤n
n
i! n k−i!1
C
C
A
xn−kyk

2n
k!=X
0≤i≤n
0≤k−i≤n
n
i! n k−i!
k0 2n k =n
2n
n!=X
0≤i≤n
0≤n−i≤n
n
i! n n−i!
=
n
X
i=0 n
i! n n−i!0≤i≤n0≤n−i≤n
=
n
X
i=0 n
i! n i! n n−i!= n i!
=
n
X
i=0 n
i!2
Pk(X)
k X `2n
n´n
2n[1,2n] 2n
2n
n!=|Pn([1,2n])|
`n
k´k n
n
k!=|Pk([1, n])|
`n
k´2k n
n
k!2
=|Pk([1, n]) × Pk([1, n])|
(A, B)k[1, n] (A, B)∈ Pk([1, n]) × Pk([1, n])
l6=k(A, B)6∈ Pl([1, n]) × Pl([1, n]) A B k l
6=lPk([1, n]) × Pk([1, n]) Pl([1, n]) × Pl([1, n])
n
X
k=0 n
k!2
=˛˛˛˛˛
n
[
k=0
Pk([1, n]) × Pk([1, n])˛˛˛˛˛
2n
n!=
n
X
k=0 n
k!2
Pn([1,2n]) Sn
k=0 Pk([1, n]) × Pk([1, n])
A[1,2n]nPn([1,2n]) A1A2
A A n
A1={p∈A p ≤n}=A∩[1, n], A2={p∈A p > n}=A∩[n+ 1,2n]

k A1A n A2n−k A0
2
A2−n A2−n
A0
2={p−n, p ∈A2}
A0
2n−k A2⊂[n+ 1,2n]A0 2⊂[1, n]B1
A0
2[1, n]B1k
A[1,2n] (A1, B1) [1, n]
|A1|=|B1|=k k A n
ϕ:Pn([1, n]) →
n
[
k=0
Pk([1, n]) × Pk([1, n])
A→(A∩[1, n],[1, n]\(A∩[n+ 1,2n]−n))
ψ:
n
[
k=0
Pk([1, n]) × Pk([1, n]) →Pn([1,2n])
(A, B)→A∪(([1, n]\B) + n)