Let \(\mathfrak{G}_m\) be the symmetric group of permutations of \(m\) symbols, let \(t(m,k)\) be the number of elements in \(\mathfrak{G}_m\) with \(k\) peaks, and \(T(m,k)\) be the number of elements in \(\mathfrak{G}_m\) with \(k\) descents of length greater than \(1\). There are recursion relations that compute \(t(m,k)\) or \(T(m,k)\) in time exponential in \(m\) and \(k\). We give closed expressions for \(t(m,k)\) and \(T(m,k)\) that can be computed in time polynomial in \(m\).

Let \([q]=\{0,\ldots,q-1\}\), let \(\Delta[q]\) denote the simplex of probability measures on \([q]\), and let \(\gamma\) denote the Lebesgue measure normalized on \(\Delta[q]\). We prove that for any symmetric monotone function \(f\colon [q]^n \to [q]\) and any \(a \in [q]\) we have \[\gamma(\{\mu \in \Delta[q]\;\vert\;\mathbf{Pr}_{x \sim \mu^{\otimes n}}[f(x)=a] \in (\varepsilon,1-\varepsilon)\}) = O(1/\log n)\textrm{.}\] We also show that this bound is tight. This improves Kalai and Mossel’s previous bound of \(O(\log\log n/\log n)\) and answers their question completely.

The Gauss circle problem asks for an approximation to the number of lattice points of \(\mathbb{Z}^2\) contained in \(B_r\), the disk of radius \(r\) centered at the origin. Upper, lower, and average bounds have been established for this number-theoretic problem and have been generalized to any lattice in any dimension. We extend this problem to a more general class of structures known as Fourier quasicrystals. Recent work from Alon, Kummer, Kurasov, and Vinzant provides an upper bound of the form \(\#(\Lambda \cap B_r) = c_0\textrm{Vol}_d(B_r) + O (r^{d-1})\) for any Fourier quasicrystal \(\Lambda \subset \mathbb{R}^d\) of density \(c_0\), where \(B_r\) is the \(d\)-dimensional ball of radius \(r\). In this paper, we improve the upper bound for any Fourier quasicrystal, by showing we can write \(\# (\Lambda \cap B_r ) = c_0\textrm{Vol}_d(B_r ) + O \left(r^{\theta(\Lambda)}\right)\), where \((d-1)/2 < \theta(\Lambda) < d-1\) is some exponent depending only on the dimension \(d\) and the growth rate of the spectrum \(S\) of \(\Lambda\). In the special case \(d = 2\), we also prove lower and upper bounds for the average of the error.

We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of