We study statistical properties of the random variables Xσ(π), the number of occurrences of the pattern σ in the permutation π. We present two contrasting approaches to this problem: traditional probability theory and the "less traditional" computational approach. Through the perspective of the first one, we prove that for any pair of patterns σ and τ , the random variables Xσ and Xτ are jointly asymptotically normal (when the permutation is chosen from Sn). 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.

We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r = 0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n = 31. We also extend those functional equations to account for the number of inversions and derive analogous algorithms.

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