Patterns in random permutations (Q2663415)

The distribution of the \(k\)!-dimensional vector of pattern densities in large random permutations of size \(n\) was studied by \textit{S. Janson} et al. [J. Comb. 6, No. 1--2, 117--143 (2015; Zbl 1312.05011)]. Their analysis showed that some component of this vector is asymptotically multi-normal of order \(1/\sqrt{n}\), while the orthogonal component is smaller. In this paper the author refines the analysis of this distribution using representations of the symmetric group and the theory of \(U\)-statistics, by showing that it decomposes into \(k\) asymptotically uncorrelated components of different orders in \(n\), that correspond to \(S_k\)-representations. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.