An involution is a permutation that is its own inverse. Given a permutation \(\sigma\) of \([n]\), let \(N_n(\sigma)\) denote the number of ways to write \(\sigma\) as a product of two involutions of \([n]\). If we endow the symmetric groups \(S_n\) with uniform probability measures, then the random variables \(N_n\) are asymptotically lognormal.

The proof is based upon the observation that, for most permutations \(\sigma\), \(N_n(\sigma)\) can be well-approximated by \(B_n(\sigma)\), the product of the cycle lengths of \(\sigma\). Asymptotic lognormality of \(N_n\) can therefore be deduced from Erdős and Turán’s theorem that \(B_n\) is itself asymptotically lognormal.

The Stirling number of the second kind \( S(n, k) \) counts the number of ways to partition a set of \( n \) labeled balls into \( k \) non-empty unlabeled cells. We extend this problem and give a new statement of the \( r \)-Stirling numbers of the second kind and \( r \)-Bell numbers. We also introduce the \( r \)-mixed Stirling number of the second kind and \( r \)-mixed Bell numbers. As an application of our results we obtain a formula for the number of ways to write an integer \( m > 0 \) in the form \( m_1 \cdot m_2 \cdot \cdots \cdot m_k \), where \( k \geq 1 \) and \( m_i \)’s are positive integers greater than 1.

Packing patterns in permutations concerns finding the permutation with the maximum number of a prescribed pattern. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently the packing density for all but two (up to equivalence) patterns up to length 4 can be obtained. In this note we consider the analogous question for colored patterns and permutations. By introducing the concept of “colored blocks” we characterize the optimal permutations with the maximum number of a given colored pattern when it contains at most three colored blocks. As examples we apply this characterization to find the optimal permutations of various colored patterns and subsequently obtain their corresponding packing densities.

We extend the main result of the paper “Arithmetic progressions in sets of fractional dimension” ([12]) in two ways. Recall that in [12], Łaba and Pramanik proved that any measure \( \mu \) with Hausdorff dimension \( \alpha \in (1 – \epsilon_0, 1) \) (here \( \epsilon_0 \) is a small constant) large enough depending on its Fourier dimension \( \beta \in (2/3, \alpha] \) contains in its support three-term arithmetic progressions (3APs). In the present paper, we adapt an approach introduced by Green in “Roth’s Theorem in the Primes” to both lower the requirement on \( \beta \) to \( \beta > 1/2 \) (and \( \epsilon_0 \) to \( 1/10 \)) and perhaps more interestingly, extend the result to show for any \( \delta > 0 \), if \( \alpha \) is large enough depending on \( \delta \), then \( \mu \) gives positive measure to the (basepoints of the) non-trivial 3APs contained within any set \( A \) for which \( \mu(A) > \delta \).

Let \( f(x) = 1 + \sum_{n=1}^\infty a_n x^n \) be a formal power series with complex coefficients. Let \( \{r_n\}_{n=1}^\infty \) be a sequence of nonzero integers. The Integer Power Product Expansion of \( f(x) \), denoted ZPPE, is \( \prod_{k=1}^\infty (1 + w_k x^k)^{r_k} \). Integer Power Product Expansions enumerate partitions of multi-sets. The coefficients \( \{w_k\}_{k=1}^\infty \) themselves possess interesting algebraic structure. This algebraic structure provides a lower bound for the radius of convergence of the ZPPE and provides an asymptotic bound for the weights associated with the multi-sets.

We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group \( \mathbb{F}_2^n \). A triangle in \( \mathbb{F}_2^n \) is a triple \( (x, y, z) \) such that \( x + y + z = 0 \). The triangle removal lemma for \( \mathbb{F}_2^n \) states that for every \( \varepsilon > 0 \) there is a \( \delta > 0 \), such that if a subset \( A \) of \( \mathbb{F}_2^n \) requires the removal of at least \( \varepsilon \cdot 2^n \) elements to make it triangle-free, then it must contain at least \( \delta \cdot 2^{2n} \) triangles.

This problem was first studied by Green [Gre05] who proved a lower bound on \( \delta \) using an arithmetic regularity lemma. Regularity-based lower bounds for triangle removal in graphs were recently improved by Fox, and we give a direct proof of an analogous improvement for triangle removal in \( \mathbb{F}_2^n \).

The improved lower bound was already known to follow (for triangle-removal in all groups) using Fox’s removal lemma for directed cycles and a reduction by Král, Serra, and Vena~\cite{KSV09} (see [Fox11, CF13]). The purpose of this note is to provide a direct Fourier-analytic proof for the group \( \mathbb{F}_2^n \).

A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of side length n for every n in the set. In [9] it is shown that N, the set of all natural numbers, tiles the plane. We answer here a number of questions from that paper. We show that there is a simple tiling of the plane (no nontrivial subset of squares forms a rectangle). We show that neither the odd numbers nor the prime numbers tile the plane. We show that N can tile many, even infinitely many planes.

Let \( s, t \) be any numbers in \( \{0,1\} \) and let \( \pi = \pi_1 \pi_2 \cdots \pi_m \) be any word. We say that \( i \in [m-1] \) is an \( (s,t) \)-parity-rise if \( \pi_i \equiv s \pmod{2} \), \( \pi_{i+1} \equiv t \pmod{2} \), and \( \pi_i < \pi_{i+1} \). We denote the number of occurrences of \( (s,t) \)-parity-rises in \( \pi \) by \( \text{rise}_{s,t}(\pi) \). Also, we denote the total sizes of the \( (s,t) \)-parity-rises in \( \pi \) by \( \text{size}_{s,t}(\pi) \), that is, \( \text{size}_{s,t}(\pi) = \sum_{\pi_i < \pi_{i+1}} (\pi_{i+1} – \pi_i). \) A composition \( \pi = \pi_1 \pi_2 \cdots \pi_m \) of a positive integer \( n \) is an ordered collection of one or more positive integers whose sum is \( n \). The number of summands, namely \( m \), is called the number of parts of \( \pi \). In this paper, by using tools of linear algebra, we found the generating function that counts the number of all compositions of \( n \) with \( m \) parts according to the statistics \( \text{rise}_{s,t} \) and \( \text{size}_{s,t} \), for all \( s, t \).

In the paper, utilizing respectively the induction, a generating function of the Lah numbers, the Chu-Vandermonde summation formula, an inversion formula, the Gauss hypergeometric series, and two generating functions of Stirling numbers of the first kind, the authors collect and provide six proofs for an identity of the Lah numbers.

We prove that if \( A \subset \mathbb{Z}_q \setminus \{0\} \), \( A \neq \langle p \rangle \), \( q = p^\ell \), \( \ell \geq 2 \) with \( |A| > C \sqrt[3]{\sqrt{\ell}^2 q^{(1-\frac{1}{4\ell})}} \), then
\[
|P(A) \cdot P(A)| \geq C’ q^3
\]
where
\[
P(A) = \left\{ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \in SL_2(\mathbb{Z}_q) : a_{11} \in A \cap \mathbb{Z}_q^\times, a_{12}, a_{21} \in A \right\}.
\]

The proof relies on a result in \([4]\) previously established by D. Covert, A. Iosevich, and J. Pakianathan, which implies that if \( |A| \) is much larger than \( \sqrt{\ell} q^{(1-\frac{1}{4\ell})} \), then
\[
|\{(a_{11}, a_{12}, a_{21}, a_{22}) \in A \times A \times A \times A : a_{11} a_{22} + a_{12} a_{21} = t\}| = |A|^4 q^{-1} + \mathcal{R}(t)
\]
where \( |\mathcal{R}(t)| \leq \ell |A|^2 q^{(1-\frac{1}{2\ell})} \).