Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. 1.

The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.

Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1.

Abstract. Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with “maximal staircases ” removed from some of its vertices. The case of one vertex corresponds to Proctor’s problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose the problem of finding a formula for the number of tilings in this case.

Abstract. We compute the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the central rhombus and the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the ‘almost central ’ rhombus above the centre.

Abstract. This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and on-line literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. 1.

We compute the number of rhombus tilings of a hexagon with sides N;M;N; N;M;N , which contain a fixed rhombus on the symmetry axis. A special case solves a problem posed by Jim Propp.

( ( x+y+j x+y+j Abstract. We evaluate the determinant det1≤i,j≤n − x−i+2j x+i+2j, which gives the number of lozenge tilings of a hexagon with cut off corners. A particularly interesting feature of this evaluation is that it requires the proof of a certain hypergeometric identity which we accomplish by using Gosper’s algorithm in a non-automatic fashion. The purpose of this paper is to provide a direct evaluation of the determinant

: This document is built around a list of twenty problems in enumeration of matchings that were gathered together in 1996 and presented in a lecture at MSRI that fall. Since then, roughly half of the problems have been solved by participants in the MSRI Workshop on Combinatorics, their students, and others. The article begins with a capsule history of the topic of enumeration of matchings. The twenty problems themselves, with commentary, comprise the bulk of the article. The final section gives an account of the progress that has been made on these problems as of this writing, and includes pointers to both the printed and on-line literature. 1 Introduction How many perfect matchings does a given graph G have? That is, in how many ways can one choose a subset of the edges of G so that each vertex of G belongs to one and only one chosen edge? (See Figure 1(a) for an example of a matching of a graph. The book by Lov'asz and Plummer [LP] gives general background on matchings of graphs.) ...

We state, discuss, provide evidence for, and prove in special cases the conjecture that the probability that a random tiling by rhombi of a hexagon with side lengths 2n+a; 2n+b; 2n+c; 2n+a; 2n+b; 2n+c contains the (horizontal) rhombus with coordinates (2n + x; 2n + y) is equal to 1 3 + g a;b;c;x;y (n) 2n n 3 . 6n 3n , where g a;b;c;x;y (n) is a rational function in n. Several specific instances of this "1/3-phenomenon" are made explicit.