Motivated by the Monthly problem #11515, we prove further interesting formulae for trigonometric series by means of telescoping method.

The main theorem establishes the generating function \(F\) which counts the number of times the staircase \(1 + 2 + 3 + \cdots + m^+\) fits inside an integer composition of \(n\).
\[
F = \frac{k_m – \frac{q x^m y}{1-x} k_{m-1}}{(1-q)x^{\binom{m+1}{2}} \left( \frac{y}{1-x} \right)^m + \frac{1-x-xy}{1-x} \left( k_m – \frac{q x^m y}{1-x} k_{m-1} \right)}.
\]
where
\[
k_m = \sum_{j=0}^{m-1} x^{mj – \binom{j}{2}} \left( \frac{y}{1-x} \right)^j.
\]

Here \(x\) and \(y\) respectively track the composition size and number of parts, whilst \(q\) tracks the number of such staircases contained.

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 \( \mathbb{F}_2^n \).