this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to e#ciently compute certain di#erential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivations on an algebra, one expects to find a Hopf algebra of some sort. We were pleased to find that not only was there a Hopf algebra associated with these data structures, but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees. Previous work applying Hopf algebras to combinatorial objects (such as [10], [13] or [14]) has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets, rather than on trees themselves. # The first author is a National Science Foundation Postdoctoral Research Fellow

A connection between the algebra of rooted trees used in renormalization theory and Runge-Kutta methods is pointed out. Butcher’s group and B-series are shown to provide a suitable framework for renormalizing a toy model of field theory, following Kreimer’s approach. Finally B-series are used to solve a class of non-linear partial differential equations. 1

We consider the fundamental operation of applying a compound filtering condition to a set of records. With large main memories available cheaply, systems may choose to keep the data entirely in main memory, in order to improve query and/or update performance. The design of a data-intensive algorithm in main memory needs to take into account the architectural characteristics of modern processors, just as a disk-based method needs to consider the physical characteristics of disk devices. An important architectural feature that influences the performance of main memory algorithms is the branch misprediction penalty. We demonstrate that branch misprediction has a substantial impact on the performance of an algorithm for applying selection conditions. We describe a space of “query plans ” that are logically equivalent, but differ in terms of performance due to variations in their branch prediction behavior. We propose a cost model that takes branch prediction into account, and develop a query optimization algorithm that chooses a plan with optimal estimated cost for conjunctive conditions. We also develop an efficient heuristic optimization algorithm. We also show how records can be ordered to further reduce branch misprediction effects.

Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically. A family of polynomials F0(x),F1(x),F2(x),... forms a convolution family if Fn(x) has degree ≤ n and if the convolution condition Fn(x + y) = Fn(x)F0(y) + Fn−1(x)F1(y) + · · · + F1(x)Fn−1(y) + F0(x)Fn(y) holds for all x and y and for all n ≥ 0. Many such families are known, and they appear frequently in applications. For example, we can let Fn(x) = x n /n!; the condition (x + y) n n! n∑ k=0 x k k! y n−k (n − k)! is equivalent to the binomial theorem for integer exponents. Or we can let Fn(x) be the binomial coefficient () x n; the corresponding identity ( ) n∑ x + y x y n k n − k is commonly called Vandermonde’s convolution. k=0 How special is the convolution condition? Mathematica will readily find all sequences of polynomials that work for, say, 0 ≤ n ≤ 4: F[n_,x_]:=Sum[f[n,j]x^j,{j,0,n}]/n! conv[n_]:=LogicalExpand[Series[F[n,x+y],{x,0,n},{y,0,n}]

Multicast services can be provided either as a basic network service or as an application-layer service. Higher level multicast implementations often provide more sophisticated features, and can provide multicast services at places where no network layer support is available. Overlay multicast networks offer an intermediate option, potentially combining the flexibility and advanced features of application layer multicast with the greater efficiency of network layer multicast. In this paper, we introduce the multicast routing problem specific to the overlay network environment and the related capacity assignment problem for overlay network planning. Our main contributions are the design of several routing algorithms that optimize the end-to-end delay and the interface bandwidth usage at the multicast service nodes within the overlay network. The interface bandwidth is typically a key resource for an overlay network provider, and needs to be carefully managed in order to maximize the number of users that can be served. Through simulations, we evaluate the performance of these algorithms under various traffic conditions and on various network topologies. The results show that our approach is cost-effective and robust under traffic variations.

: The Narayana distribution is N n;k = 1 n i n k\Gamma1 ji n k j for 1 k n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining the n th Narayana polynomial as N n (w) = P 1kn N n;k w k , for n 1, the paper gives a combinatorial proof of a three term recurrence for these polynomials. It examines the Schroder numbers and the Kirkman numbers and establishes a sequence of bijections linking dissections of polygons to large Schroder paths. keywords: Catalan numbers, lattice paths, Schroder numbers AMS Subject Classification: 05A15 email: sulanke@math.idbsu.edu Note to typesetter: the symbol L is a calligraphic L. It should be set as either a calligraphic L or a script L. Thanks. 1 2 1. introduction Using the steps V = (0; 1) and H = (1; 0), the set of Catalan paths, Cat n , is the set of lattice paths from (0; ...

Introduction to Combinatorial Analysis, Wiley, NY, 1958. [46] J. Riordan, Combinatorial Identities, Wiley, NY, 1968. [47] J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83--93. Reprinted in C. E. Shannon, Collected Ppaers, edited N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993. [48] F. S. Roberts, Applied. Combinatorics, Prentice-Hall, Englewood Cliffs, NJ, 1984. [49] D. G. Rogers, A Schroder triangle: three combinatorial problems, in Combinatorial Mathematics V (Melbourne 1976), ed. C. H. C. Little, Lecture Notes in math., Vol. 622 (1977), pp. 175--196. [50] G.-C. Rota, The number of partitions of a set, Amer. Math. Monthly, 71 (1964), 498--504. [51] G.-C. Rota et al., Finite Operator Calculus, Academic Press, NY, 1975. [52] M. Schader,

Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute q-Wick products and normal products in terms of each other. 1.

We present here a survey of most notable Delannoy's works. These works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the rst general way to solve Ballot-like problems. We also give a tentative short biography.

This paper considers combinatorial interpretations for two triangular recurrence arrays containing the Schroder numbers, s n = 1; 1; 3; 11; 45; ... and r n = 1; 2; 6; 22; 90; ... , for n = 0; 1; 2; .... These interpretations involve the enumeration of constrained lattice paths and bicolored parallelogram polyominoes, called zebras. In addition to two recent inductive constructions of zebras and their associated generating trees, we present two new ones and a bijection between zebras and constrained lattice paths. We use the constructions with generating functions methods to count sets of zebras with respect to natural parameters.