grandes-ecoles 2024 Q3

grandes-ecoles · France · x-ens-maths-a__mp Permutations & Arrangements Combinatorial Structures on Permutation Matrices/Groups

Establish that $$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_n : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_n : \varepsilon(\sigma) = -1\right\}$$ and deduce the probability that a permutation of $\mathfrak{S}_n$ drawn uniformly at random has a prescribed signature.

Establish that
$$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_n : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_n : \varepsilon(\sigma) = -1\right\}$$
and deduce the probability that a permutation of $\mathfrak{S}_n$ drawn uniformly at random has a prescribed signature.

Paper Questions

Q1a Q1b Q2 Q3 Q4 Q5a Q5b Q5c Q5d Q6 Q7a Q7b Q8a Q8b Q8c Q9 Q10 Q11 Q12 Q13a Q13b Q14a Q14b Q15 Q16 Q17a Q17b Q17c Q17d Q18 Q19a Q19b Q19c Q19d Q20a Q20b Q20c Q21a Q21b Q22a Q22b Q22c Q22d Q23