benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Let S={1,2,…,n} for some integer n>1. Say a permutation π of S has a …k∈S if …π(k)>π(k+1) for k=1; …π(k−1)<π(k) and π(k)>π(k+1) for 1<k<n; …π(k−1)<π(k) for k=n. …n=5 and π takes va…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let S={1,2,…,n}S = \{1, 2, \dots, n\} for some integer n>1n > 1. Say a permutation π\pi of SS has a \emph{local maximum} at k∈Sk \in S if \begin{enumerate} \item[(i)] π(k)>π(k+1)\pi(k) > \pi(k+1) for k=1k=1; \item[(ii)] π(k−1)<π(k)\pi(k-1) < \pi(k) and π(k)>π(k+1)\pi(k) > \pi(k+1) for 1<k<n1 < k < n; \item[(iii)] π(k−1)<π(k)\pi(k-1) < \pi(k) for k=nk=n. \end{enumerate} (For example, if n=5n=5 and π\pi takes values at 1,2,3,4,51, 2, 3, 4, 5 of 2,1,4,5,32, 1, 4, 5, 3, then π\pi has a local maximum of 2 at k=1k=1, and a local maximum of 5 at k=4k=4.) What is the average number of local maxima of a permutation of SS, averaging over all permutations of SS?
Plain-text mathematical notation (without MathML)
Let S={1,2,…,n} for some integer n>1. Say a permutation π of S has a \emph{local maximum} at k∈S if \begin{enumerate} \item[(i)] π(k)>π(k+1) for k=1; \item[(ii)] π(k−1)<π(k) and π(k)>π(k+1) for 1<k<n; \item[(iii)] π(k−1)<π(k) for k=n. \end{enumerate} (For example, if n=5 and π takes values at 1,2,3,4,5 of 2,1,4,5,3, then π has a local maximum of 2 at k=1, and a local maximum of 5 at k=4.) What is the average number of local maxima of a permutation of S, averaging over all permutations of S?
Original LaTeX notation
Let $S = \{1, 2, \dots, n\}$ for some integer $n > 1$. Say a permutation $\pi$ of $S$ has a \emph{local maximum} at $k \in S$ if \begin{enumerate} \item[(i)] $\pi(k) > \pi(k+1)$ for $k=1$; \item[(ii)] $\pi(k-1) < \pi(k)$ and $\pi(k) > \pi(k+1)$ for $1 < k < n$; \item[(iii)] $\pi(k-1) < \pi(k)$ for $k=n$. \end{enumerate} (For example, if $n=5$ and $\pi$ takes values at $1, 2, 3, 4, 5$ of $2, 1, 4, 5, 3$, then $\pi$ has a local maximum of 2 at $k=1$, and a local maximum of 5 at $k=4$.) What is the average number of local maxima of a permutation of $S$, averaging over all permutations of $S$?

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
54fcb147-dbcd-5516-9b8e-f087da1743c1
task_key
test--54fcb147-dbcd-5516-9b8e-f087da1743c1
task_revision_id
1
config / split
/ test

Official source

initial import