grandes-ecoles 2019 Q46

grandes-ecoles · France · centrale-maths2__psi Sequences and Series Functional Equations and Identities via Series

For $j \in \mathbb{N}$, we denote by $Y_{n,j}$ the set of partitions whose first term $\alpha_1$ is less than or equal to $j$ and by $y_{n,j}$ the cardinality of $Y_{n,j}$; we set $y_{0,0} = 1$.
We propose to show that, if $2 \leqslant j \leqslant n$, then $y_{n,j} = y_{n,j-1} + y_{n-j,\min(j,n-j)}$.
For $j < n$, verify that $y_{n,j} = y_{n,j-1} + y_{n-j,j}$. Conclude.

For $j \in \mathbb{N}$, we denote by $Y_{n,j}$ the set of partitions whose first term $\alpha_1$ is less than or equal to $j$ and by $y_{n,j}$ the cardinality of $Y_{n,j}$; we set $y_{0,0} = 1$.

We propose to show that, if $2 \leqslant j \leqslant n$, then $y_{n,j} = y_{n,j-1} + y_{n-j,\min(j,n-j)}$.

For $j < n$, verify that $y_{n,j} = y_{n,j-1} + y_{n-j,j}$. Conclude.

Paper Questions

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