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pg5001:fonctions_de_couplage - Wiki Jeux et Mathématiques
Wiki Jeux et Mathématiques

Ensembles Partie II - 2^(n-2) bijections de N^n vers N - Partie III - (n-1)! bijections de N^n vers N

Fonctions de couplage de Skolem

Les fonctions de couplages sont des bijections de $ \mathbb N^n $ vers $ \mathbb N $.
Les fonctions présentées dans cette page sont des polynômes de degré $ n $ en $ x = (x_1, x_2, \cdots , x_n) $, c'est-à-dire en $ n $ variables $x_1, x_2,\cdots, x_n $. Lorsque $ n = 2 $, on retrouve le polynôme de Cantor qui est une bijection de $ \mathbb N^2 $ vers $ \mathbb N $, dans le cas particulier $ n=2 $.
Toute permutation $ \sigma $ des variables $ x_1, x_2, \cdots , x_n $ permet d'obtenir une nouvelle fonction de couplage $ P $ $ \circ $ $ \sigma $, si $ P $ est un polynôme de degré $ n $, alors $ P $ $ \circ $ $ \sigma $ est aussi un polynôme de degré $ n $.
La première image représente la fonction de Cantor, l'image dans $ \mathbb N $ est indiquée en face du point de coordonnées $ (x_1, x_2) $ de $ \mathbb N^2 $.


La deuxième image représente la fonction de Skolem de $ \mathbb N^3 $ vers $ \mathbb N $, seules quelques valeurs sont indiquées sur le schéma de gauche.



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