On the limiting distribution for the length of the longest alternating sequence in a random permutation
The electronic journal of combinatorics, Tome 13 (2006)
Recently Richard Stanley initiated a study of the distribution of the length as$_n(w)$ of the longest alternating subsequence in a random permutation $w$ from the symmetric group ${\cal S}_n$. Among other things he found an explicit formula for the generating function (on $n$ and $k$) for Pr$\,$(as$_n(w)\le k)$ and conjectured that the distribution, suitably centered and normalized, tended to a Gaussian with variance 8/45. In this note we present a proof of the conjecture based on the generating function.
@article{10_37236_1051,
author = {Harold Widom},
title = {On the limiting distribution for the length of the longest alternating sequence in a random permutation},
journal = {The electronic journal of combinatorics},
year = {2006},
volume = {13},
doi = {10.37236/1051},
zbl = {1086.05012},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1051/}
}
Harold Widom. On the limiting distribution for the length of the longest alternating sequence in a random permutation. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1051
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