Dan Romik - Local extrema in random permutations and the structure of longest alternating subsequences

dmtcs:2956 - Discrete Mathematics & Theoretical Computer Science, January 1, 2011, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) - https://doi.org/10.46298/dmtcs.2956
Local extrema in random permutations and the structure of longest alternating subsequencesArticle

Authors: Dan Romik 1

  • 1 Department of Mathematics [Univ California Davis]

Let $\textbf{as}_n$ denote the length of a longest alternating subsequence in a uniformly random permutation of order $n$. Stanley studied the distribution of $\textbf{as}_n$ using algebraic methods, and showed in particular that $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ and $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. From Stanley's result it can be shown that after rescaling, $\textbf{as}_n$ converges in the limit to the Gaussian distribution. In this extended abstract we present a new approach to the study of $\textbf{as}_n$ by relating it to the sequence of local extrema of a random permutation, which is shown to form a "canonical'' longest alternating subsequence. Using this connection we reprove the abovementioned results in a more probabilistic and transparent way. We also study the distribution of the values of the local minima and maxima, and prove that in the limit the joint distribution of successive minimum-maximum pairs converges to the two-dimensional distribution whose density function is given by $f(s,t) = 3(1-s)t e^{t-s}$.


Volume: DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
Section: Proceedings
Published on: January 1, 2011
Imported on: January 31, 2017
Keywords: longest alternating subsequences,permutation statistics,random permutation,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • CAREER: Combinatorial probability, limit shapes and enumeration; Funder: National Science Foundation; Code: 0955584

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