
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
where
Here
An involution is a permutation that is its own inverse. Given a permutation
The proof is based upon the observation that, for most permutations
The Stirling number of the second kind
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
Let
We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group
This problem was first studied by Green [Gre05] who proved a lower bound on
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
1970-2025 CP (Manitoba, Canada) unless otherwise stated.