test de pdf

∀p, q ∈N,[[p, q]] = [p, q]∩N
n∈N[[1, n]]
E F E F E
n∈N[[1, n]] E
N
E|E|#ECardE
Card∅= 0 E A ⊂E A |A|6|E|
E E
E∩F=∅=⇒ |E∪F|=|E|+|F|
E∩F=∅EtF
Fi∈IAi =Pi∈I|Ai|
E E
E
A⊂E|E\A|=|E|−|A| |E∪F|+|E∩F|=|E|+|F|
|E×F|=|E|·|F|
Qi∈IAi =Qi∈I|Ai| |An|=|A|n
FEE F P(E)
EPk(E)k
FE =|F||E|
χAA E
E→ {0,1}χA(x) = 1 x∈A χA(x) = 0
A7→ χAP(E){0,1}E|P(E)|= 2|E|

E F E F
∀x, y ∈F, |f−1(x)|=|f−1(y)|k
F|E|=k· |F|
E F m n
E F n(n−1) . . . (n−m+ 1) m=n
E F n!
n(n−1) . . . (n−m+ 1) m n (n)m
k n
k n k [[1, n]]
k n k [[1, n]]
k n k [[1, n]]
k n k [[1, n]]
Ak
n
n
kCk
nn
k
k n
(n)kn
k
nkn
k=n−k+ 1 k
•n
k= 0 k < 0k > n
•n
k=n n−k
•kn
k=nn−1 k−1
•X
kn
k= 2n
•n
k=n−1 k−1+n−1 k
•X
kp
k q n−k=p+q nX
kn
k2
=2n
n
A E
ENA E
A A

∀x, y, (x+y)n=X
kn
kxkyn−k
n k {a1, . . . , ak}ni
ai
n
n1n−n1 n2. . . nk−1+nk nk−1=n! n1!n2!. . . nk!
n
n1, n2, . . . , nk
∀x1, . . . , xk,(x1+x2+· · · +xk)n=X
n(n1,n2,...,nk)∈N∗
n1+n2+···+nk=n
n
n1, n2, . . . , nk·xn1
1xn2 2· · · xnk
k

n∈N[[1, n]]
E E
n n n
SnS(E)
|Sn|=n!
σ∈Snσ=1 2 . . . n
σ(1) σ(2) . . . σ(n)
E
(x, σ(x))
[[1, n]]
σ σ =
σ(1) σ(2) . . . σ(n) [[1, n]]
Sn
n σ
σ◦σ−1=σ−1◦σ= id(= 1 2 3 . . . n)
σ, τ ∈Snστ =σ◦τ στ(i) = σ(τ(i))
σ=31425 σ−1=24135
σ i σ(i) = i σ
σ
Supp(σ)
σ τ Supp(σ)∩Supp(τ) = ∅στ =τσ
σ=2341
τ=2134 στ =3241 τσ =1342
i σ i σ {sk(i)|k∈N}
σ i σ
σ[[1, n]]
i k > 0
σk(i) = i
` `

γ∈Sn` i
γ{i, γ(i), . . . , γ`(i)}γ= (i γ(i). . . γ`(i))
i ` γ
σ= 2 3 4 1 = (1 2 3 4) = (2 3 4 1) = (3 4 1 2) = (4 1 2 3)
σ= 4 3 5 1 2 = (1 4) (2 3 5) = (3 5 2) (4 1) = . . .
n k c(n, k)c(0,0) = 1
c(n, k) = (n−1)c(n−1, k) + c(n−1, k −1)
2
Snn
2(n−1)!
`n
`(`−1)! = (n)`
`
Sn
(i, j)σ∈Sni < j σ−1(i)< σ−1(j)
σ=. . . i . . . j . . .
i < j σ(i)> σ(j)
I(σ)I(σ)σ
˜σ σ ˜σ(i) = σ(i)
I(σ) = I(σ−1I(σ) + I(˜σ) = n
2
1
4n(n−1)
σ∈Snn c(σ) = (c1(σ), . . . , cn(σ))
∀i∈[[1, n]] ci(σ) = #{j > i|σ(j)< σ(i)}i
σ n t(σ) =
(t1(σ), . . . , tn(σ)) ∀i∈[[1, n]] ti(σ) = #{j < σ−1(i)|σ(j)> i}
i i
t(σ) = c(σ−1)c(σ) = t(σ−1)