grandes-ecoles 2017 QIIA

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

We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Show that for $n \geqslant 1$, $B _ { n }$ equals the total number of partitions of the set $\llbracket 1 , n \rrbracket$.

We set for every integer $n \geqslant 0$,
$$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$
where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.

Show that for $n \geqslant 1$, $B _ { n }$ equals the total number of partitions of the set $\llbracket 1 , n \rrbracket$.

Paper Questions

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