Exercice - Alain Camanes

2015/2016
Exercice 1. (Décomposition en base factorielle, ! )
1. n∈N?n
P
k=1
k·k! = (n+ 1)! −1
2. N∈N?n∈N?n
(a1, . . . , an)∈Nnk∈N?06ak6k an6= 0 N=
n
P
k=1
ak·k!
Exercice 2. (Identité de Vandermonde, -)x∈R, n, p ∈N
1. (1 + X)n(1 + X)p
2. k6inf(n, p)n+p
k=
k
P
j=0 n
k−jp j
Exercice 3. (Le même exercice, un autre point de vue, -)E n
A, B E p q A =c B
1. f:P(E)→P(A)×P(B), X 7→ (A∩X, B ∩X)
2. r∈N
r
P
i=0 p
iq r−i=p+q r
3. n
P
i=0 n
i2
Exercice 4. ( ! )p6n
p
P
k=0 n
kn−k p−k
Exercice 5. k n k 6n
k(i1, . . . , ik)J1, nKk
1.
2.
3. 16i1<··· < ik6n
Exercice 6. n n1
n2p
p1p2
1.
2.
3.
Exercice 7. (-)5 10
1. 0
2.

Exercice 8. (Géométrie) n∈N?n
Exercice 9. n
Exercice 10.
1. 2.
Exercice 11. ( ! )n, p ∈N?p(u1, . . . , up)∈Np
p
P
k=1
uk=n
Exercice 12. n∈N?E n
(X, Y )∈P(E)2X⊂Y
Exercice 13. (n, p)∈(N?)2J1, pK
J1, nK
Exercice 14. (Relations binaires, ♥)n∈N?E n
E
1.
2.
3.
4.
Exercice 15. (Surjections, Projections) n∈N?E=J1, nK
1. J1, n + 1K J1, nK
2. E E p ◦p=p
Exercice 16. (Surjections) Sp
nJ1, pK J1, nKSp n
1. k∈J1, nKAk={f∈F(E, F ) ; f−1({k}) = ∅}
2.
Sp
n=
n
X
k=0
(−1)n−kn
kkp.
Exercice 17. (Dérangements, ! ,♥)n∈N?, n >2Dn
{1, . . . , n}f:{1, . . . , n} → {1, . . . , n}
k6n f(k)6=k
Exercice 18. (Signature, -)
1. 12345678
35487621.2. 12345678 13274856.

3. 1 2 3 4 ··· 2n−1 2n
n n + 1 n−1n+ 2 ··· 1 2n.
Exercice 19. (-)σ θi,j J1, nK
1. σ◦(i, j)◦σ−1= (σ(i), σ(j))
2. n>3Sn
Exercice 20. (♥)Sn
1. {(1, i), i ∈J2, nK}
2. {(i, i + 1), i ∈J1, n −1K}
3. (1,2) (2,3,··· , n, 1)
Exercice 21. (♥)
3An3
Exercice 22. ( ! )c= (2,3,··· , n, 1) {σ∈Sn;σc =cσ}
Exercice 23. ϕ: (Sn,◦)→(C?,·)
1. θ1θ2ϕ(θ1) = ϕ(θ2)
2. ϕ1