On the Limiting Distribution for the Longest Alternating Sequence in a Random Permutation

Recently Richard Stanley [3] initiated a study of the distribution of the length asn(w) of the longest alternating subsequence in a random permutation w from the symmetric group Sn. Among other things he found an explicit formula for the generating function (on n and k) for Pr (asn(w) ≤ 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. If w = (w1 · · ·wn) is a permutation in the symmetric group Sn then an increasing subsequence of length k is a subsequence wi1 · · ·wik satisfying wi1 < wi2 · · · < wik . The random variable isn(w), the length of the longest increasing sequence in a random permuation w from Sn, has been much studied. Its mean was first determined asymptotically by Logan-Shepp [2] and Vershik-Kerov [4], proving a conjecture of Ulam, and the limiting distribution was determined in the celebrated work of Beik-Deift-Johansson [1]. Recently Stanley [3] initiated a study of the distribution of the length of the longest alternating subsequences of w, a subsequence such that wi1 > wi2 < wi3 > · · ·wik . If asn(w) denotes the length of the longest alternating sequence in a random permutation w from Sn denote by pn(k) the distribution function for asn(w), pn(k) = 1 n! {#w : asn(w) ≤ k}.