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Papers (191)

2025

centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38

2024

centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26

2023

centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15

2022

centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23

2021

centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44

2020

centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18

2019

centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26

2018

centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24

2017

centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26

2016

centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20

2015

centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22

2014

centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24

2013

centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9

2012

centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18

2011

centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10

2010

centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27

2019 centrale-maths2__psi

49 maths questions

Q1 Matrices Linear Transformation and Endomorphism Properties View

What can be said about a nilpotent endomorphism of index 1?

Q2 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Show that there exists a vector $x$ in $E$ such that $u^{p-1}(x) \neq 0$.

Q3 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Verify that the family $\left(u^{k}(x)\right)_{0 \leqslant k \leqslant p-1}$ is free. Deduce that $p = 2$.

Q4 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Show that $\operatorname{Ker} u = \operatorname{Im} u$.

Q5 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Construct a basis of $E$ in which the matrix of $u$ is equal to $J_2$.

Q6 Matrices Eigenvalue and Characteristic Polynomial Analysis View

We assume that $n = 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Deduce that the nilpotent matrices in $\mathcal{M}_2(\mathbb{C})$ are exactly the matrices with zero trace and zero determinant.

Q7 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Show that $\operatorname{Im} u \subset \operatorname{Ker} u$ and that $2r \leqslant n$.

Q8 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Assume that $\operatorname{Im} u = \operatorname{Ker} u$. Show that there exist vectors $e_1, e_2, \ldots, e_r$ in $E$ such that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right)\right)$ is a basis of $E$.

Q9 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$. Assume that $\operatorname{Im} u = \operatorname{Ker} u$ and that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right)\right)$ is a basis of $E$.
Give the matrix of $u$ in this basis.

Q10 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Assume $\operatorname{Im} u \neq \operatorname{Ker} u$. Show that there exist vectors $e_1, e_2, \ldots, e_r$ in $E$ and vectors $v_1, v_2, \ldots, v_{n-2r}$ belonging to $\operatorname{Ker} u$ such that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right), v_1, \ldots, v_{n-2r}\right)$ is a basis of $E$.

Q11 Matrices Linear Transformation and Endomorphism Properties View

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$. Assume $\operatorname{Im} u \neq \operatorname{Ker} u$ and that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right), v_1, \ldots, v_{n-2r}\right)$ is a basis of $E$.
What is the matrix of $u$ in this basis?

Q12 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that, if $A$ is nilpotent, then 0 is the unique eigenvalue of $A$.

Q13 Matrices Diagonalizability and Similarity View

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
What are the matrices in $\mathcal{M}_n(\mathbb{C})$ that are both nilpotent and diagonalizable?

Q14 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that a matrix is nilpotent if, and only if, its characteristic polynomial is equal to $X^n$.

Q15 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show the converse of question 12: if 0 is the unique eigenvalue of $A$, then $A$ is nilpotent.

Q16 Matrices Diagonalizability and Similarity View

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that an upper triangular matrix in $\mathcal{M}_n(\mathbb{C})$ with zero diagonal is nilpotent and that a nilpotent matrix is similar to an upper triangular matrix with zero diagonal.

Q17 Matrices Linear Transformation and Endomorphism Properties View

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Prove that, if $A$ is a nilpotent matrix of index $p$, then every polynomial in $\mathbb{C}[X]$ that is a multiple of $X^p$ is an annihilating polynomial of $A$.

Q18 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$. Assume that $P$ is an annihilating polynomial of $A$ nilpotent.
Prove that 0 is a root of $P$.

Q19 Matrices Linear System and Inverse Existence View

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$. Assume that $P$ is an annihilating polynomial of $A$ nilpotent.
We denote by $m$ the multiplicity of 0 in $P$, which allows us to write $P = X^m Q$ where $Q$ is a polynomial in $\mathbb{C}[X]$ such that $Q(0) \neq 0$. Prove that $Q(A)$ is invertible and then that $P$ is a multiple of $X^p$ in $\mathbb{C}[X]$.

Q20 Matrices Determinant and Rank Computation View

We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$.
Calculate the trace and rank of $A$. Deduce, without any calculation, the characteristic polynomial of $A$. Show that $A$ is nilpotent and give its nilpotency index.

Q21 Matrices Diagonalizability and Similarity View

We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$.
Prove that $A$ is similar to the matrix $\operatorname{diag}\left(J_2, J_1\right)$. Give the value of an invertible matrix $P$ such that $A = P \operatorname{diag}\left(J_2, J_1\right) P^{-1}$.

Q22 Matrices Linear Transformation and Endomorphism Properties View

We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$. We seek to determine the set of matrices $R \in \mathcal{M}_3(\mathbb{C})$ such that $R^2 = A$. We denote by $\rho$ the endomorphism canonically associated with $R$.
Prove that $\operatorname{Im} u$ and $\operatorname{Ker} u$ are stable under $\rho$ and that $\rho$ is nilpotent.

Q23 Matrices Linear Transformation and Endomorphism Properties View

We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$. We seek to determine the set of matrices $R \in \mathcal{M}_3(\mathbb{C})$ such that $R^2 = A$. We denote by $\rho$ the endomorphism canonically associated with $R$.
Deduce the set of square roots of $A$. One may consider $R' = P^{-1}RP$.

Q24 Matrices Matrix Power Computation and Application View

We propose to study the matrix equation $R^2 = J_3$.
Let $R$ be a solution of this equation. Give the values of $R^4$ and $R^6$, then the set of solutions of the equation.

Q25 Matrices Determinant and Rank Computation View

Let $V \in \mathcal{M}_n(\mathbb{C})$ be a nilpotent matrix of index $p$. We propose to study the equation $R^2 = V$.
Show that, if $2p - 1 > n$, then there is no solution.

Q26 Matrices Linear Transformation and Endomorphism Properties View

Let $V \in \mathcal{M}_n(\mathbb{C})$ be a nilpotent matrix of index $p$. We propose to study the equation $R^2 = V$.
For every value of the integer $n \geqslant 3$, exhibit a matrix $V \in \mathcal{M}_n(\mathbb{C})$, nilpotent of index $p \geqslant 2$ and admitting at least one square root.

Q27 Matrices Linear Transformation and Endomorphism Properties View

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Prove that $\operatorname{Im} u$ is stable under $u$ and that the endomorphism induced by $u$ on $\operatorname{Im} u$ is nilpotent. Specify its nilpotency index.

Q28 Matrices Linear Transformation and Endomorphism Properties View

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
For every non-zero vector $x$ in $E$, we denote by $C_u(x)$ the vector space spanned by the $\left(u^k(x)\right)_{k \in \mathbb{N}}$; prove that $C_u(x)$ is stable under $u$ and that there exists a smallest integer $s(x) \geqslant 1$ such that $u^{s(x)}(x) = 0$.

Q29 Matrices Linear Transformation and Endomorphism Properties View

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$. For every non-zero vector $x$ in $E$, we denote by $C_u(x)$ the vector space spanned by the $\left(u^k(x)\right)_{k \in \mathbb{N}}$ and $s(x)$ the smallest integer $\geqslant 1$ such that $u^{s(x)}(x) = 0$.
Prove that $(x, u(x), \ldots, u^{s(x)-1}(x))$ is a basis of $C_u(x)$ and give the matrix, in this basis, of the endomorphism induced by $u$ on $C_u(x)$.

Q30 Matrices Linear Transformation and Endomorphism Properties View

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Prove by induction on $p$ that there exist vectors $x_1, \ldots, x_t$ in $E$ such that $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$. One may apply the induction hypothesis to the endomorphism induced by $u$ on $\operatorname{Im}(u)$.

Q31 Matrices Linear Transformation and Endomorphism Properties View

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$, and suppose there exist vectors $x_1, \ldots, x_t$ in $E$ such that $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$.
Give the matrix of $u$ in a basis adapted to the decomposition $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$.

Q32 Matrices Diagonalizability and Similarity View

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$.
Show that there exist a partition $\sigma = \left(\alpha_1, \ldots, \alpha_k\right)$ of $n$ and a basis $\mathcal{B}$ of $E$ in which the matrix of $u$ is equal to the matrix $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$.

Q33 Matrices Determinant and Rank Computation View

Let $\alpha$ be a non-zero natural integer. Calculate the rank of $J_\alpha^j$ for every natural integer $j$. Deduce that $J_\alpha$ is nilpotent and specify its nilpotency index.

Q34 Matrices Determinant and Rank Computation View

Q35 Matrices Determinant and Rank Computation View

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For $j \in \mathbb{N}$, we denote $\Lambda_j = \left\{i \in \llbracket 1, k \rrbracket \mid \alpha_i \geqslant j\right\}$. Prove that $\operatorname{rg}\left(N_\sigma^j\right) = \sum_{i \in \Lambda_j} \left(\alpha_i - j\right)$.

Q36 Matrices Determinant and Rank Computation View

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Prove that, for every $j \in \mathbb{N}^*$, the integer $d_j = \operatorname{rg}\left(u^{j-1}\right) - \operatorname{rg}\left(u^j\right)$ is equal to the number of blocks $J_{\alpha_i}$ whose size $\alpha_i$ is greater than or equal to $j$.

Q37 Matrices Determinant and Rank Computation View

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Give the value of the integer $k$, the number of blocks $J_{\alpha_i}$ appearing in $N_\sigma$.

Q38 Matrices Determinant and Rank Computation View

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For every integer $j$ between 1 and $n$, express the number of blocks $J_{\alpha_i}$ of size exactly equal to $j$.

Q39 Matrices Diagonalizability and Similarity View

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis $\mathcal{B}$ is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Assume that there exist a partition $\sigma'$ of the integer $n$ and a basis $\mathcal{B}'$ of $E$ such that the matrix of $u$ in $\mathcal{B}'$ is equal to $N_{\sigma'}$. Show that $\sigma = \sigma'$.

Q40 Matrices Structured Matrix Characterization View

What is the maximum cardinality of a set of nilpotent matrices, all of the same size $n$, such that there are no two similar matrices in this set?

Q41 Matrices Diagonalizability and Similarity View

Let $A$ be the matrix $\left(\begin{array}{ccccc} 0 & -1 & 2 & -2 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & -1 & 1 & 0 \end{array}\right)$ and $u$ the endomorphism canonically associated with $A$.
Determine the partition $\sigma$ of the integer 5 associated with $u$ and give the matrix $N_\sigma$.

Q42 Matrices Diagonalizability and Similarity View

Using the result of question 31, prove that if $M \in \mathcal{M}_n(\mathbb{C})$ is nilpotent, then $M$, $2M$ and $M^\top$ are similar.

Q43 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Using the result of question 15, prove that if $M$ and $2M$ are similar, then $M$ is nilpotent.

Q44 Sequences and Series Evaluation of a Finite or Infinite Sum View

Q45 Sequences and Series Functional Equations and Identities via Series View

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 $y_{n,j} = y_{n,j-1} + y_{n-j,\min(j,n-j)}$.
Prove that this equality is true for $j = n$.

Q46 Sequences and Series Functional Equations and Identities via Series View

Q47 Sequences and Series Evaluation of a Finite or Infinite Sum View

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 $y_{0,0} = 1$.
Calculate the $y_{n,j}$ for $1 \leqslant j \leqslant n \leqslant 5$ by presenting the results in the form of a table.

Q48 Sequences and Series Algorithmic/Computational Implementation for Sequences and Series View

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 $y_{0,0} = 1$.
Write a Python function that takes as argument an integer $n \geqslant 1$ and returns $y_{n,n}$.

Q49 Sequences and Series Evaluation of a Finite or Infinite Sum View

Compare the result of question 48 (the value of $y_{n,n}$, the number of partitions of $n$) to that of question 40 (the maximum cardinality of a set of pairwise non-similar nilpotent matrices of size $n$).