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.

. We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)-enumeration of these lozenge tilings. In the case that a = b = c, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For m = 0, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det 0i;jn\Gamma1 \Gamma !ffi ij + \Gamma m+i+j j \Delta\Delta , w...

. We give a short, self-contained evaluation of the Andrews--Burge determinant (Pacific J. Math. 158 (1994), 1--14). 1. Introduction In [9, Theorem 1], Andrews and Burge proved a determinant evaluation equivalent to (1.1) det 0i;jn\Gamma1 " x + i + j 2i \Gamma j ' + ` y + i + j 2i \Gamma j " = (\Gamma1) (nj3 mod 4) 2 ( n 2 )+1 \Theta n\Gamma1 Y j=1 \Gamma x+y 2 + j + 1 \Delta b(j+1)=2c \Gamma \Gamma x+y 2 \Gamma 3n + j + 3 2 \Delta bj=2c (j) j ; where the shifted factorial (a) k is given by (a) k := a(a + 1) \Delta \Delta \Delta (a + k \Gamma 1), k 1, (a) 0 := 1, and where (A)=1 if A is true and (A)=0 otherwise. This determinant identity arose in connection with the enumeration of symmetry classes of plane partitions. The known proofs [9, 10] of (1.1) require that one knows (1.1) to hold for x = y. Indeed, the latter was first established by Mills, Robbins and Rumsey [15, p. 53], in turn using another determinant evaluation, due to Andrews [3], whose p...

Propp conjectured [15] that the number of lozenge tilings of a semiregular hexagon of sides 2n \Gamma 1, 2n \Gamma 1 and 2n which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregular hexagon of sides a, a and b. We prove explicit formulas for the number of lozenge tilings of these hexagons containing the central unit rhombus, and obtain Propp's conjecture as a corollary of our results.

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.

We prove q-analogues of two determinant identities of a previous paper of the author. These determinant identities are related to the enumeration of totally symmetric self-complementary plane partitions.

. We evaluate the determinant det 0i;jn\Gamma1 i ffi ij + P n\Gamma1 t;k=0 \Gamma i+ t \Delta\Gamma k+ k\Gammat \Delta \Gamma j \Gammak+\Gamma1 j \Gammak \Delta 2 k\Gammat j which gives the 2-enumeration of certain shifted plane partitions. This generalizes a result of Andrews (Aequationes Math. 33 (1987), 230--250), who evaluated this determinant for = 0, thereby proving a conjecture of Mills, Robbins and Rumsey (Discrete Math. 67 (1987), 43--55). 1. Introduction. In their investigation on plane partition enumeration, Mills, Robbins and Rumsey [11, p. 50] came across the determinant det 0i;jn\Gamma1 / ffi ij + n\Gamma1 X t=0 n\Gamma1 X k=0 ` i + t '` k k \Gamma t '` j \Gamma k + \Gamma 1 j \Gamma k ' x k\Gammat ! ; (1.1) which (in sec. 4 of their paper) they identified as the generating function for certain shifted plane partitions. For x = 1, this determinant was evaluated by Andrews [1]. The case = 0 established the enumeration of cyclically symme...

Abstract. A determinant evaluation is proven, a special case of which establishes a conjecture of Bombieri, Hunt, and van der Poorten (Experimental Math. 4 (1995), 87–96) that arose in the study of Thue’s method of approximating algebraic numbers. 1. Introduction. In their study [2] of Thue’s method of approximating an algebraic number, Bombieri, Hunt, and van der Poorten conjectured two determinant evaluations, one of which can be restated as follows. Conjecture (Bombieri, Hunt, van der Poorten [2, next-to-last paragraph]). Let b, c be nonnegative integers, c ≤ b, and let ∆(b, c) be the determinant of the