Algorithmique. Memento de combinatoire 1. Quelques définitions 2

E F f :E→F
E∀(x, y)∈
E×E, f (x) = f(y)⇒x=y
F∀y∈F, ∃x∈E|f(x) = y
f
E F E ≈F
f:E→F a b [a, b] := {n∈
N|a≤n≤b}E n
E E ≈[1, n] #E|E|E
u:E→F E =NE u
F uk:= u(k) (u(k))k∈Eu
I E I
(Ei)i∈IEi:= E(i)
E F
n= #E p = #F
•E×F#(E×F) = n.p
•E→F#FE=pn
•E#P(E) = 2n
•E→F n ≤p An
p:= p!
(p−n)!
•E→E#Sn=n! := n(n−1)(n−2) . . . 1
•E p Cp
n=n
p:= n! (n−p)!p!
•n p Rp
n:= (p+n−1)!
(p−1)!
•Rp
n
n!
•E Bn+1 :=
n
k=0 n
kBk, B0= 1.
•E p
Sp
n+1 := Sp−1
n+pSp n=
n
k=0 n
kSp−1
k, S1
n=Sn n= 1
•E F p!Sp
n
E F
f:E→F
P(n)Na∈NP(n)
P(a)
∀n∈N, P (n)⇒P(n+ 1)
P(n)n≥a
•(Ei)i∈IE
#E=
i∈I
#Ei.
•Sn:= n
i=0 ui
unq̸= 1
Sn=(n+ 1)(nq + 2u0)
2(Sn=u0(qn+1 −1) q−1).
•p n 0≤p≤n
n
p=n n−p=n−1 p+n−1 p−1n
p=0 n
p= 2n.
•a, b
(a+b)n=
n
p=0 n
papbn−p.
•n p
[n/p] [x]x
•E F f :E→F
F k #E=k#F
•q̸= 1 n
n
k=0
kqk=q
(q−1)2[qn(n(q−1) + 1) + 1]
n
p=0
pn
p=n2n−1