Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.

. A signicant portion of MacMahon's famous book \Combinatory Analysis " is devoted to the development of \Partition Analysis" as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. Nevertheless, MacMahon's ideas have not received due attention with the exception of work by Richard Stanley. A long range object of a series of articles is to change this situation by demonstrating the power of MacMahon's method in current combinatorial and partition-theoretic research. The renaissance of MacMahon's technique partly is due to the fact that it is ideally suited for being supplemented by modern computer algebra methods. In this paper we illustrate the use of Partition Analysis and of the corresponding package Omega by focusing on three dierent aspects of combinatorial work: the construction of bijections (for the Rened Lecture Hall Partition Theorem), exploitation of recursive patterns (for Cayley composit...

In his famous book "Combinatory Analysis" MacMahon introduced Partition Analysis as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. The object of this paper is to introduce an entirely new application domain for MacMahon's operator technique. Namely, we show that Partition Analysis can be also used for proving hypergeometric multisum identities. Our examples range from combinatorial sums involving binomial coefficients, harmonic and derangement numbers to multisums which arise in physics and which are related to the Knuth-Bender theorem.

In his famous book "Combinatory Analysis" MacMahon introduced Partition Analysis ("Omega Calculus") as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. The objective of this paper is to show that partition analysis is ideally suited for being supplemented by modern computer algebra methods. We developed the computer algebra package Omega which implements various aspects of MacMahon's ideas. In addition to an introduction to basic facts of "Omega Calculus", we present a variety of applications that illustrate the usage of the package. 1 Introduction Recall the beautiful refinement of Euler's classic result [1, p. 5] that was discovered by M. Bousquet-M'elou and K. Eriksson [5] only recently: Partially supported by the visiting researcher program of the J. Kepler University Linz. y Supported by SFB-grant F1305 of the Austrian FWF. 1 Theorem 1 ("Lecture Hall Partition Theorem"). The number of p...

We present a unified theory for permutation models of all the infinite families of finite and a#ne Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott's formula (in the refined version of Macdonald) for the Poincareseriesofthese a#ne Weyl groups. 1991 Mathematics Subject Classification. primary 20B35; secondary 05A15. 1 Introduction The aim of this paper is to present a unified theory for permutation representations of the finite Weyl groups A n-1 , B n , C n , D n , and the a#ne Weyl groups # A n-1 , # B n , # C n , # D n . Our starting point is the symmetric group S n , the group of permutations of [1,...,n]. If S n is presented as the group generated by adjacent transpositions, it is isomorphic to the Weyl group A n-1 , and we obtain well-known interpretations of several Coxeter group concepts in permutation language: 1. The Coxeter generators are the adjacent transpositions. 2. Reflections corresp...

Abstract. Two q-analogues of Euler’s theorem on integer partitions with odd or distinct parts are given. A q-lecture hall theorem is given. Euler’s Odd=Distinct theorem is 1.

This paper examines percolation questions in a deterministic setting. In particular, I consider R, the set of elements of Z² with greatest common divisor equal to 1, where two sites are connected if they are at distance 1. The main result of the paper proves that the infinite component has an asymptotic density. An “almost everywhere” sieve of J. Friedlander is used to obtain the result.