We consider the problem of packing fixed-length patterns into a permutation, and develop a connection between the number of large patterns and the number of bonds in a permutation. Improving upon a result of Kaplansky and Wolfowitz, we obtain exact values for the expectation and variance for the number of large patterns in a random permutation. Finally, we are able to generalize the idea of bonds to obtain results on fixed-length patterns of any size, and present a construction that maximizes the number of patterns of a fixed size.

We present in this work results on some distributions of permutation statistics of random elements of the wreath product \( G_{r,n} = C_r \wr S_n \). We consider the distribution of the descent number, the flag major index, the excedance, and the number of fixed points, over the whole group \( G_{r,n} \), or over the subclasses of derangements and involutions. We compute the mean, variance and moment generating function, and establish the asymptotic distributions of these statistics.

Let \( A \) be a subset of \( \mathbb{F}_p^n \), the \( n \)-dimensional linear space over the prime field \( \mathbb{F}_p \), of size at least \( \delta N \) (\( N = p^n \)), and let \( S_v = P^{-1}(v) \) be the level set of a homogeneous polynomial map \( P : \mathbb{F}_p^n \to \mathbb{F}_p^R \) of degree \( d \), for \( v \in \mathbb{F}_p^R \). We show that, under appropriate conditions, the set \( A \) contains at least \( c N|S| \) arithmetic progressions of length \( l \leq d \) with common difference in \( S_v \), where \( c \) is a positive constant depending on \( \delta \), \( l \), and \( P \). We also show that the conditions are generic for a class of sparse algebraic sets of density \( \approx N^{-\gamma} \).