We study the Euler–Frobenius numbers, a generalization of the Eulerian numbers, and the probability distribution obtained by normalizing them. This distribution can be obtained by rounding a sum of independent uniform random variables; this is more or less implicit in various results and we try to explain this and various connections to other areas of mathematics, such as spline theory.
The mean, variance and (some) higher cumulants of the distribution are calculated. Asymptotic results are given. We include a couple of applications to rounding errors and election methods.

An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a collection of height comparison inequalities for heights over a number field and over a quaternion algebra. We also show how these inequalities can be used to obtain existence results for points of bounded height over a quaternion algebra, which constitute non-commutative analogues of variations of the classical Siegel’s lemma and Cassels’ theorem on small zeros of quadratic forms.

We prove that when a pre-independence space satisfies some natural properties, then its cyclic flats form a bounded lattice under set inclusion. Additionally, we show that a bounded lattice is isomorphic to the lattice of cyclic flats of a pre-independence space. We also prove that the notion of cyclic width gives rise to dual-closed and minorclosed classes of B-matroids. Finally, we find a difference between finite matroids and B-matroids by using the notion of well-quasi-ordering.

In this paper, we generalize an earlier statistic on square-and-domino tilings by considering only those squares covering a multiple of k, where k is a fixed positive integer. We consider the distribution of this statistic jointly with the one that records the number of dominos in a tiling. We derive both finite and infinite sum expressions for the corresponding joint distribution polynomials, the first of which reduces when k = 1 to a prior result. The cases q = 0 and q = −1 are noted for general k. Finally, the case k = 2 is considered specifically, where further results may be given, including a combinatorial proof when q = −1.

A sequence of coefficients appearing in a recurrence for the Narayana polynomials is generalized. The coefficients are given a probabilistic interpretation in terms of beta distributed random variables. The recurrence established by M. Lasalle is then obtained from a classical convolution identity. Some arithmetical properties of the generalized coefficients are also established.

We consider the problem of packing fixed-length patterns into a permutation, and develop a connection between the number of large patterns and the number of bonds in a permutation. Improving upon a result of Kaplansky and Wolfowitz, we obtain exact values for the expectation and variance for the number of large patterns in a random permutation. Finally, we are able to generalize the idea of bonds to obtain results on fixed-length patterns of any size, and present a construction that maximizes the number of patterns of a fixed size.

We present in this work results on some distributions of permutation statistics of random elements of the wreath product \( G_{r,n} = C_r \wr S_n \). We consider the distribution of the descent number, the flag major index, the excedance, and the number of fixed points, over the whole group \( G_{r,n} \), or over the subclasses of derangements and involutions. We compute the mean, variance and moment generating function, and establish the asymptotic distributions of these statistics.

Let \( A \) be a subset of \( \mathbb{F}_p^n \), the \( n \)-dimensional linear space over the prime field \( \mathbb{F}_p \), of size at least \( \delta N \) (\( N = p^n \)), and let \( S_v = P^{-1}(v) \) be the level set of a homogeneous polynomial map \( P : \mathbb{F}_p^n \to \mathbb{F}_p^R \) of degree \( d \), for \( v \in \mathbb{F}_p^R \). We show that, under appropriate conditions, the set \( A \) contains at least \( c N|S| \) arithmetic progressions of length \( l \leq d \) with common difference in \( S_v \), where \( c \) is a positive constant depending on \( \delta \), \( l \), and \( P \). We also show that the conditions are generic for a class of sparse algebraic sets of density \( \approx N^{-\gamma} \).

In this paper, we investigate properties of a new class of generalized Cauchy numbers. By using the method of coecient, we establish a series of identities involving generalized Cauchy numbers, which generalize some results for the Cauchy numbers. Furthermore, we give some asymptotic approximations of certain sums related to the generalized Cauchy numbers.

Let \( F(x) = \sum_{\nu \in \mathbb{N}^d} F_\nu x^\nu \) be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume \( F = G / H \) for some functions \( G \) and \( H \) holomorphic in a neighborhood of the origin. We derive asymptotics for the coefficients \( F_{r\alpha} \) as \( r \to \infty \) with \( r\alpha \in \mathbb{N}^d \) for \( \alpha \) in a permissible subset of \( d \)-tuples of positive reals. More specifically, we give an algorithm for computing arbitrary terms of the asymptotic expansion for \( F_{r\alpha} \) when the asymptotics are controlled by a transverse multiple point of the analytic variety \( H = 0 \). This improves upon earlier work by R. Pemantle and M. C. Wilson. We have implemented our algorithm in Sage and apply it to obtain accurate numerical results for several rational combinatorial generating functions.