Feuille 2 - Research page of Charles Grellois

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{1, . . . , n} {1, . . . , m}
n, m ∈NFn
m{1, . . . , n}
{1, . . . , m}
Fn
m
Tn
mn
{1,2, . . . , m}
f∈Fn
mTf∈Tn m1≤i≤n Tf[i] = f(i)
f7→ TfFn
mTn m
Fn
m
n m
n1m
1 0
n
m
n
mTn m
n
m
T T n
m
Tn
mn mn m
T
Tn
m
Tn
mn m
??
σ(i, j) (i, j)i<j
σ(i)> σ(j) (σ(i), σ(j)) σ
•3 2 4 1
σ
i i = 1, . . . , n
•6321547
•n
σ(1) {1,2,3,4,5,6,7}σ(1) = 6
σ(2) {1,2,3,4,5,7}
σ(2) = 3

•
•
•n
•
n
σ{1,...,n}
i∈ {1,...,n}σ(σ(i)) = i i ∈ {1,...,n}σ(i)6=i
σ{1,...,n}S S[i−1] = σ(i)
i∈ {1,...,n}
{1,...,4}
[1,2,3,4] [2,1,4,3] [4,3,2,1] [1,3,2,4]
n>0S
σ{1, . . . , n}1σ0
n>0
{1,...,n}
{1,...,n}n
?
n
σ{1,...,n}k k
ka1, . . . , ak{1,...,n}σ(ai) = ai+1 i∈ {1,...,k−1}
σ(ak) = a1j /∈ {a1, . . . , an}σ(j) = j
k∈ {2, . . . , n}k
σ{1,...,n}S S[i−1] = σ(i)
i∈ {1,...,n}
{1,2,3,4}
n>0 2
{1, . . . , n}
3
n>0k>0
?
k{1, . . . , n}
n>0k∈ {2, . . . , n}
??
Sσ{1, . . . , n}1σ
k0
n>0Sσ{1, . . . , n}1
σ0