grandes-ecoles 2017 QIB

grandes-ecoles · France · centrale-maths1__pc Permutations & Arrangements Combinatorial Proof or Identity Derivation

Throughout the problem, for every pair $( n , k )$ of strictly positive integers, we denote by $S ( n , k )$ the number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts. We further set $S ( 0,0 ) = 1$ and, for all $( n , k ) \in \mathbb { N } ^ { * 2 } , S ( n , 0 ) = S ( 0 , k ) = 0$.
Express $S ( n , k )$ as a function of $n$ or of $k$ in the following cases:
I.B.1) $k > n$;
I.B.2) $k = 1$.

Throughout the problem, for every pair $( n , k )$ of strictly positive integers, we denote by $S ( n , k )$ the number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts. We further set $S ( 0,0 ) = 1$ and, for all $( n , k ) \in \mathbb { N } ^ { * 2 } , S ( n , 0 ) = S ( 0 , k ) = 0$.

Express $S ( n , k )$ as a function of $n$ or of $k$ in the following cases:

I.B.1) $k > n$;

I.B.2) $k = 1$.

Paper Questions

QIA QIB QIC QID QIIA QIIB QIIC QIID QIIE QIIF QIIIA QIIIB QIIIC QIIID QIVA QIVB QIVC QVA QVB QVC QVD QVE