These are partitions of [l] = {1, 2, . . . , l} into n blocks such that no four term subsequence of [l] induces the mentioned pattern and each k consecutive numbers of [l] fall into different blocks. These structures are motivated by Davenport-Schinzel sequences. We summarize and extend known enumerative results for the pattern p = abab and give an explicit formula for the number p(abab, n, l, ...

This paper will prove that essentially only the obvious recurrences have almost all primes as divisors. An integer n is a divisor of a recurrence if n divides some term of the recurrence. In this paper, "almost all primes" will be taken interchangeably to mean either all but finitely many primes or all but for a set of Dirichlet density zero in the set of primes. In the context of this paper, t...

Numbers of the form Fb,n = b 2n+1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2m−1. The theoretical distributions of GFN primes, for fixed n, are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Ba...

A set of integers A is called a B2[g] set if every integer m has at most g representations of the form m = a+ a′, with a a′ and a, a′ ∈ A. We obtain a new lower bound for F(g, n), the largest cardinality of a B2[g] set in {1, . . . , n}. More precisely, we prove that lim infn→∞ F(g,n) √gn 2 π − εg where εg → 0 when g → ∞. We show a connection between this problem and another one discussed by Sc...

In the theory of generalized Davenport–Schinzel sequences one estimates the maximum lengths of finite sequences containing no subsequence of a given pattern. Here we investigate a further generalization, in which the class of sequences is extended to the class of colored trees. We determine exactly the extremal functions associated with the properly 2-colored path of four vertices and with the ...

An extremal problem considering sequences related to Davenport-Schinzel sequences is investigated in this paper. We prove that f(x1x i 2..x i kx i 1x i 2..x i k, n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a1a2..am of integers such that 1) 1 ≤ ar ≤ n, 2) ar = as, r 6= s implies |r− s| ≥ k and 3) u contains no subsequence of the type x1......

Consider a face F in an arrangement of n Jordan curves in the plane, no two of which intersect more than s times. We prove that the combinatorial complexity of F is O(λs(n)), O(λs+1(n)), and O(λs+2(n)), when the curves are bi-infinite, semi-infinite, or bounded, respectively; λk(n) is the maximum length of a Davenport-Schinzel sequence of order k on an alphabet of n symbols. Our bounds asymptot...