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 \( \mu(A) > \delta \).