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# Journal of Combinatorics

## Volume 6 (2015)

### Number 1–2

### On the asymptotic statistics of the number of occurrences of multiple permutation patterns

Pages: 117 – 143

DOI: http://dx.doi.org/10.4310/JOC.2015.v6.n1.a8

#### Authors

#### Abstract

We study statistical properties of the random variables $X_{\sigma} (\pi)$, the number of occurrences of the pattern $\sigma$ in the permutation $\pi$. We present two contrasting approaches to this problem: traditional probability theory and the “less traditional” computational approach. Through the perspective of the first approach, we prove that for any pair of patterns $\sigma$ and $\tau$, the random variables $X_{\sigma}$ and $X_{\tau}$ are jointly asymptotically normal (when the permutation is chosen from $S_n$). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normality (up to a point) and derive explicit formulas for quite a few moments and mixed moments empirically, yet rigorously. The computational approach can also be extended to the case where permutations are drawn from a set of pattern avoiders to produce many empirical moments and mixed moments. This data suggests that some random variables are not asymptotically normal in this setting.

#### Keywords

permutation pattern, random permutation, joint asymptotic normality, mixed moments