his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_

Many applications compute aggregate functions...

Abstract not found

High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.

Mathematical notation evolves like all languages do. As new experiments are made, we sometimes witness the survival of the fittest, sometimes the survival of the most familiar. A healthy conservatism keeps things from changing too rapidly; a healthy radicalism keeps things in tune with new theoretical emphases. Our mathematical language continues to improve, just as “the d-ism of Leibniz overtook the dotage of Newton ” in past centuries [4, Chapter 4]. In 1970 I began teaching a class at Stanford University entitled Concrete Mathematics. The students and I studied how to manipulate formulas in continuous and discrete mathematics, and the problems we investigated were often inspired by new developments in computer science. As the years went by we began to see that a few changes in notational traditions would greatly facilitate our work. The notes from that class have recently been published in a book [15], and as I wrote the final drafts of that book I learned to my surprise that two of the notations we had been using were considerably more useful than I had previously realized. The ideas “clicked ” so well, in fact, that I’ve decided to write this article, blatantly attempting to promote these notations among the mathematicians who have no use for [15]. I hope that within five years everybody will be able to use these notations in published papers without needing to explain what they mean.

the best known answer is Faà di Bruno’s Formula. If g and f are functions with a sufficient number of derivatives, then dm g ( f (t)) = dtm � m! b1! b2!···bm! g(k) � � ′ b1 � ′ ′ f (t) f (t) ( f (t))

Starting with divided differences of binomial coefficients, a class of multivalued polynomials (three parameters), which includes Bernoulli and Stirling polynomials and various generalizations, is developed. These carry a natural and convenient combinatorial interpretation. Some particular calculations are done and several factorization results are proven and conjectured. 1. Introduction Our study began with a binomial identity involving an alternating sum of vector space dimensions, which arose in the course of proving Bezout's Theorem. The identity led to consideration of a class of polynomials, which are best understood as higher order divided differences of binomial coefficients. These polynomials are closely related to many of the standard polynomials of combinatorial analysis, in particular to the Stirling polynomials, which determine their coefficients. Although it was not possible to find a closed form in general, which was the original intention, calculation of particular va...

integrals. Abstract. The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulæ for the distribution of the sum of n non-identically distributed uniform random variables in both the continuous and the discrete case. The results, though involved, have a certain elegance. As examples, we derive from our general formulæ some special cases which have appeared in the literature.

Abstract. In this paper we compute the 2-adic valuations of some polynomials associated with the definite integral ∫ ∞ dx 0 (x4 + 2ax2. + 1) m+1 1.

A bivariate symmetric backwards recursion is of the form d[m, n]=w 0 (d[m1, n]+d[m, n-1])+# 1 (d[m-r 1 ,n-s 1 ]+d[m-s 1 ,n-r 1 ])++# k (d[m-r k ,n-s k ] +d[m-s k ,n-r k ]) where # 0 ,...# k are weights, r 1 ,...r k and s 1 ,...s k are positive integers. We prove three theorems about solving symmetric backwards recursions restricted to the diagonal band x + u<y<x-l. With a solution we mean a formula that expresses d[m, n] as a sum of di#erences of recursions without the band restriction. Depending on the application, the boundary conditions can take di#erent forms. The three theorems solve the following cases: d[x+u, x] = 0 for all x # 0, and d[x- l, x] = 0 for all x # l (applies to the exact distribution of the Kolmogorov-Smirnov two-sample statistic), d[x + u, x]=0 for all x # 0, and d[x - l +1,x]=d[x-l+1,x-1] for x # l (ordinary lattice paths with weighted left turns), and d[y, y - u +1]=d[y-1,y-u+1]for all y # u and d[x - l +1,x]=d[x-l+1,x-1] for x # l. The first theorem is a gene...