A problem of mine appeared in the lastest Mathematics Magazine:

Suppose \(\pi\) is a permutation of \(\{1,2,\ldots,2m\}\). Consider the (possibly empty) subsequence of \(\pi(m+1),\pi(m+2),\ldots,\pi(2m)\) consisting of only those values which exceed \(\max\{\pi(1),\ldots,\pi(m)\}\).

Let \(P(m)\) denote the probability that this subsequence never decreases, when \(\pi\) is a randomly chosen permutation of \(\{1,2,\ldots,2m\}\). Evaluate the limit of \(P(m)\) as \(m\to\infty\).