Motivated by the work of Chung, Claesson, Dukes, and Graham in [5], we define a natural type B analog of the classic bubble sort, and use it to define a type B analog of the maximum drop statistic. We enumerate (by explicit, recursive, and generating function formulas) signed permutations with r type B descents and type B maximum drop at most k. We also find a connection between these signed pe...

Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1992. They have been developed by many authors since, for various combinatorial objects such as strings, subsets, permutations, partitions, vector spaces, and designs. One generalization of universal cycles, which require almost complete overlap of consecutiv...

Using the main ideas of Tanaka, the measure-solution {Pt}t of a 3-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to {Pt}t of solutions {P l t}t of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allow...

In this paper we study the packet routing problem under the matching model proposed by Alon, Chung and Graham 1]. We extend the model to allow more than one packet per origin and destination node. We give tight bounds for the many-to-one routing number for complete graphs, complete bipartite graphs and linear arrays. We also present an eecient algorithm for many-to-one routing on an trees (and ...

I have been as puzzled about this comment as Zavadsky was about my writings. What I was puzzled about was, which of my ideas or concepts were considered by Zavadsky to be so close to those of the dialectical materialists. I have been wondering about this for the past 30 years and I think I have gradually come close to an answer. In this I have been helped by a number of publications, particular...

We present a polynomial time algorithm for precedence constrained scheduling problems in which the task graph can be partitioned into large disjoint parts by removing edges with high oat, where the oat of an edge is deened as the diierence between the length of the longest path in the graph and the length of the longest path containing the edge. Our algorithm guarantees schedules within a facto...

Let s(n) be the side length of the smallest square into which n non-overlapping unit squares can be packed. We show that s(m2 − 3) = m for m = 4, 7, implying that the most efficient packings of 13 and 46 squares are the trivial ones. The study of packing unit squares into a square goes back to Erdős and Graham [2], who showed that large numbers of unit squares can be packed in a way that is sur...

An (R, S; m, n)k-de Bruijn torus is a k-ary R × S toroidal array with the property that every k-ary m × n matrix appears exactly once contiguously on the torus. The torus is a generalization of de Bruijn cycles and has been extended to higher dimensions by many authors. The central question, asked by Chung, Diaconis, and Graham, is for which R, S, m, n, and k such tori exist. In this note we de...

An old question of Erdős asks if there exists, for each number N , a finite set S of integers greater than N and residue classes r(n) (mod n) for n ∈ S whose union is Z. We prove that if ∑ n∈S 1/n is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdős and Selfridge. We also prove a conjecture of Erdős and Graham, that, f...

Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often, the generating tree for the set of permutations avoiding a set of restrictions Q requires infinitely many labels. Sometimes, however, this generating tree needs only finitely many labels. We characterize the finite sets of restrictions Q for which this ...