Christian Krattenthaler

Advanced Determinant Calculus

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.

Vicious walkers, friendly walkers and Young tableaux: II. With a wall

We derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of Young tableaux and combinatorial descriptions of symmetric functions. For the problem of n friendly walkers, we derive exact asymptotics for the number of stars and watermelons, both in the absence of a wall and in the presence of a wall.

Superconformal indices of three-dimensional theories related by mirror symmetry

Recently, Kim, and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by mirror symmetry. The proofs are based on well-known identities in the theory of q-special functions. We also suggest a general index formula taking into account the U(1)J global symmetry present for abelian theories.

A new matrix inverse

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.

Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole

We deal with 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 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 det0?i, j?n?1(??ij+(m+i+jj)), where ? is any 6th root of unity. These determinant evaluations are variations of a famous result due to Andrews (1979, Invent. Math.53, 193?225), which corresponds to ?=1.

GENERATING FUNCTIONS FOR PLANE PARTITIONS OF A GIVEN SHAPE

For fixed integers α and β, planar arrays of integers of a given shape, in which the entries decrease at least by α along rows and at least by β along columns, are considered. For various classes of these (α,β)-plane partitions we compute three different kinds of generating functions. By a combinatorial method, determinantal expressions are obtained for these generating functions. In special cases these determinants may be evaluated by a simple determinant lemma. All known results concerning plane partitions of a given shape are included. Thus our approach of a given shape provides a uniform proof method and yields numerous generalizations of known results.

Stanley decompositions and Hilbert depth in the Koszul complex

Stanley decompositions of multigraded modules

Another Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula

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The Enumeration of Lattice Paths With Respect to Their Number of Turns

over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley conjectured that the Stanley depth of a module

Automatic generation of hypergeometric identities by the beta integral method

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