Opérations dans \(\mathbb{N}^n\)

[ Definition ]

Étant donnés \(\nu\) et \(\eta\) dans \(\mathbb{N}^n\):

\(\bullet\)on note \(\nu !=\Pi_{i=1}^n (\nu_i)!\).

\(\bullet\)on note \(\nu \geq \eta\) si \(\forall i\in [1,n] \nu_i - \eta_i \geq 0\)

\(\bullet\)si \(\nu \geq \eta\) on note \(\alpha=\nu-\eta\) avec \(\forall i\in [1,n] \alpha_i = \nu_i - \eta_i\)

\(\bullet\)si \(\nu \geq \eta\) on note \(C_\nu^\eta = \frac{\nu!}{\eta ! (\nu-\eta)!}=\Pi_{i=1}^n C_{\nu_i}^{\eta_i}\)

\(\bullet\)on note \(|\nu|=\sum_{i=1}^n \nu_i\) (longueur de \(\nu\))

\(\bullet\)on note \(0\) l’élément \((0,...,0)\) de \(\mathbb{N}^n\).

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