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The purpose of this article is threefold First it provides the reader with a few useful and e cient tools which should enable her him to evaluate nontrivial de terminants 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… (More)

We compute the inverse of a specific infinite-dimensional matrix, thus unifying a number of previous matrix inversions. Our inversion theorem is applied to derive a number of summation formulas of hypergeometric type.

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… (More)

We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and nestings of matchings and set partitions in the larger context of the enumeration of fillings of Ferrers shape on which one imposes restrictions on their increasing and decreasing chains. While Chen et al. work with Robinson–Schensted-like insertion/deletion algorithms, we use the… (More)

A theory of counting nonintersecting lattice paths by the major index and generalizations of it is developed. We obtain determinantal expressions for the corresponding generating functions for families of nonintersecting lattice paths with given starting points and given nal points, where the starting points lie on a line parallel to x + y = 0. In some… (More)

- Mihai Ciucu, Theresia Eisenkölbl, Christian Krattenthaler, D. Zare
- J. Comb. Theory, Ser. A
- 2001

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 (?1)-enumeration of these lozenge tilings. In the case that a = b = c, we… (More)

We prove a formula, conjectured by Conca and Herzog, for the number of all families of nonintersecting lattice paths with certain starting and end points in a region that is bounded by an upper ladder. Thus we are able to compute explicitly the Hilbert series for certain one-sided ladder determinantal rings.

We survey old and new results on the enumeration of lattice paths in the plane with a given number of turns, including the recent developments on the enumeration of nonintersecting lattice paths with a given number of turns. Motivations to consider such enumeration problems come from various elds, e.g. probability, statistics, combinatorics, and commutative… (More)

- Mihai Ciucu, Christian Krattenthaler
- J. Comb. Theory, Ser. A
- 2002

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… (More)

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W , such that S (but not necessarily R) is reduced. For each such pair (R, S) we construct a family of W -invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters q, t1, t2, . . . , tr, where r (= 1,… (More)