For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. Let $n \in \mathbf { N }$. Show that $P _ { n , N }$ is included in $\llbracket 0 , n \rrbracket ^ { N }$ and non-empty for all $N \in \mathbf { N } ^ { * }$, that the sequence $\left( p _ { n , N } \right) _ { N \geq 1 }$ is increasing and that it is constant from rank $\max ( n , 1 )$ onwards.
For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality.
Let $n \in \mathbf { N }$. Show that $P _ { n , N }$ is included in $\llbracket 0 , n \rrbracket ^ { N }$ and non-empty for all $N \in \mathbf { N } ^ { * }$, that the sequence $\left( p _ { n , N } \right) _ { N \geq 1 }$ is increasing and that it is constant from rank $\max ( n , 1 )$ onwards.