We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

We say that a permutation σ ∈ S n contains a permutation π ∈ S k as a pattern if some subsequence of σ has the same order relations among its entries as π . We improve on results of Wilf, Coleman, and Eriksson et al. that bound the asymptotic behavior of pat(n), the maximum number of distinct patterns of any length contained in a single permutation of length n. We prove that 2 n − O (n 2 2 n− √ 2n ) pat(n) 2 n − Θ(n2 n− √ 2n ) by estimating the amount of redundancy due to patterns that are contained multiple times in a given permutation. We also consider the question of ksuperpatterns, which are permutations that contain all patterns of a given length k. We give a simple construction that shows that L k , the length of the shortest k-superpattern, is at most. This may lend evidence to a conjecture of Eriksson et al. that L k ∼ k 2 2 .

We study the dynamical Mahler measure of multivariate polynomials and present dynamical analogues of various results from the classical Mahler measure as well as examples of formulas allowing the computation of the dynamical Mahler measure in certain cases. We discuss multivariate analogues of dynamical Kronecker's Lemma and present some improvements on the result for two variables due to Carter, Lalín, Manes, Miller, and Mocz.

We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of Boyd-Lawton formula and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker's lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer's question is true.

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