Abstract: We study a generalization of the \emph{Frobenius problem}: given $k$ positive relatively prime integers, what is the largest integer $g_0$ that cannot be represented as a nonnegative integral linear combination of these parameters? More generally, what is the largest integer $g_s$ that has exactly $s$ such representations? We illustrate a family of parameters, based on a recent paper by Tripathi, whose generalized Frobenius numbers $g_0, \ g_1, \ g_2, ...$ exhibit unnatural jumps; namely, $g_0, \ g_1, \ g_k, \ g_{\binom{k+1}{k-1}}, \ g_{\binom{k+2}{k-1}}, ...$ form an arithmetic progression, and any integer larger than $g_{\binom{k+j}{k-1}}$ has at least $\binom{k+j+1}{k-}$ representations. Along the way, we introduce a variation of a generalized Frobenius number and prove some basic results about it.

Abstract: : Recently, Tu and Deng proposed a combinatorial conjecture about binary strings and, on the assumption that the conjecture is correct, they obtained two classes of Boolean functions that are both algebraic immunity optimal, the first class of which are also bent functions. The second class gives balanced functions, which have optimal algebraic degree and the best nonlinearity known until now. In this paper, using three different approaches, we prove that this conjecture is true in many cases with different counting strategies. We also propose some problems about the weight equations that are related to this conjecture. Because of the scattered distribution, we predict that an exact count will be difficult to obtain, in general.

Abstract: Let gj denote the largest integer that is represented exactly j times as a non-negative integer linear combination of fx1,...,xng. We show that for any k > 0, and n = 5, the quantity g0 − gk is unbounded. Furthermore, we provide examples with g0 > gk for n � 6 and g0 > g1 for n � 4.

Abstract: We give explicit formulae for obtaining the binary sequences which produce Steinhaus triangles and generalized Pascal triangles with rotational and dihedral symmetries.

Abstract: We show that a completely multiplicative automatic function, which does not take 0 as a value, is almost periodic.

Abstract: Abstract We reintroduce an interpretation of the kth-fold self convolution of the Catalan numbers by showing that they count the number of words in symbols X and Y, where the total number of Y's is k more than the total number of X's, and at no time are there more Y's than k plus the number of X's. Using this, we exhibit some of the wide variety of combinatorial interpretations of the kth-fold self convolution of the Catalan numbers. Finally, we show how these numbers appear as the last column in a truncated Pascal's triangle.

Abstract: The study of the limiting distribution of eigenvalues of N × N random matrices as N → ∞ has many applications, including nuclear physics, number theory and network theory. One of the most studied ensembles is that of real symmetric matrices with independent entries drawn from identically distributed nice random variables, where the limiting rescaled spectral measure is the semi-circle. Studies have also determined the limiting rescaled spectral measures for many structured ensembles, such as Toeplitz and circulant matrices. These systems have very different behavior; the limiting rescaled spectral measures for both have unbounded support. Given a structured ensemble such that (i) each random variable occurs o(N) times in each row of matrices in the ensemble and (ii) the limiting rescaled spectral measure μ exists, we introduce a parameter to continuously interpolate between these two behaviors. We fix a p ∈ [1/2, 1] and study the ensemble of signed structured matrices by multiplying the (i, j)th and (j, i)th entries of a matrix by a randomly chosen ǫij ∈ {1,−1}, with Prob(ǫij = 1) = p (i.e., the Hadamard product). For p = 1/2 we prove that the limiting signed rescaled spectral measure is the semi-circle. For all other p, we prove the limiting measure has bounded (resp., unbounded) support if μ has bounded (resp., unbounded) support, and converges to μ as p → 1. Notably, these results hold for Toeplitz and circulant matrix ensembles. The proofs are by Markov’s Method of Moments. The analysis of the 2kth moment for such distributions involves the pairings of 2k vertices on a circle. The contribution of each pairing in the signed case is weighted by a factor depending on p and the number of vertices involved in at least one crossing. These numbers are of interest in their own right, appearing in problems in combinatorics and knot theory. The number of configurations with no vertices involved in a crossing is well-studied, and are the Catalan numbers. We discover and prove similar formulas for configurations with 4, 6, 8 and 10 vertices in at least one crossing. We derive a closed-form expression for the expected value and determine the asymptotics for the variance for the number of vertices in at least one crossing. As the variance converges to 4, these results allow us to deduce properties of the limiting measure.

Abstract: We prove that if a subset of a $d$-dimensional vector space over a finite field with $q$ elements has more than $q^{d-1}$ elements, then it determines all the possible directions. If a set has more than $q^k$ elements, it determines a $k$-dimensional set of directions. We prove stronger results for sets that are sufficiently random. This result is best possible as the example of a $k$-dimensional hyperplane shows. We can view this question as an Erd\H os type problem where a sufficiently large subset of a vector space determines a large number of configurations of a given type. For discrete subsets of ${\Bbb R}^d$, this question has been previously studied by Pach, Pinchasi and Sharir.

Abstract: Motivated by the Cauchy-Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result: There is a universal constant e > 0 such that, if G is a connected, regular graph on n vertices, then either every pair of vertices can be connected by a path of length at most three, or the number of pairs of such vertices is at least 1+e times the number of edges in G. We discuss a range of further questions to which this result gives rise.

Abstract: A ternary inclusion-exclusion polynomial is a polynomial of the form \[ Q_{p,q,r}=\frac{(z^{pqr}-1)(z^p-1)(z^q-1)(z^r-1)} {(z^{pq}-1)(z^{qr}-1)(z^{rp}-1)(z-1)}, \] where $p$, $q$, and $r$ are integers $\ge3$ and relatively prime in pairs. This class of polynomials contains, as its principle subclass, the ternary cyclotomic polynomials corresponding to restricting $p$, $q$, and $r$ to be distinct odd prime numbers. Our object here is to continue the investigation of the relationship between the coefficients of $Q_{p,q,r}$ and $Q_{p,q,s}$, with $r\equiv s\pmod{pq}$. More specifically, we consider the case where $1\le s<\max(p,q)