Ilse Fischer

Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and Equipartition

We propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities. Our approach is based on Lagrangian duality, where the inequalities are dualized, and only a basic set of constraints is maintained explicitly. This leads to function evaluations requiring to solve a relatively simple semidefinite program. Our approach provides accurate solutions to semidefinite relaxations of the Max-Cut and the Equipartition problem, which are not achievable by direct approaches based only on interior-point methods.

A Characterisation of Pfaffian Near Bipartite Graphs

A graph is 1-extendible if every edge has a 1-factor containing it. A 1-extendible non-bipartite graph G is said to be near bipartite if there exist edges e1 and e2 such that G?{e1, e2} is 1-extendible and bipartite. We characterise the Pfaffian near bipartite graphs in terms of forbidden subgraphs. The theorem extends an earlier characterisation of Pfaffian bipartite graphs.

A new proof of the refined alternating sign matrix theorem

In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of nxn alternating sign matrices with a 1 at the top of the ith column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(-1) matrices generalizing alternating sign matrices.

Enumeration of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus in the Centre

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.

The operator formula for monotone triangles - simplified proof and three generalizations

We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted enumeration of monotone triangles which specializes to the weighted enumeration of alternating sign matrices with respect to the number of -1s in the matrix when prescribing (1,2,...,n) as the bottom row of the monotone triangle.

Fully Packed Loops in a triangle

Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the study of ordinary Fully Packed Loop configurations (FPLs) on the square grid where they were used to show that the number of FPLs with a given link pattern that has m nested arches is a polynomial function in m. It soon turned out that TFPLs possess a number of other nice properties. For instance, they can be seen as a generalized model of Littlewood-Richardson coefficients. We start our article by introducing oriented versions of TFPLs; their main advantage in comparison with ordinary TFPLs is that they involve only local constraints. Three main contributions are provided. First, we show that the number of ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs and thus it suffices to consider the latter. Second, we decompose oriented TFPLs into two matchings and use a classical bijection to obtain two families of nonintersecting lattice paths (path tangles). This point of view turns out to be extremely useful for giving easy proofs of previously known conditions on the boundary of TFPLs necessary for them to exist. One example is the inequality d ( u ) + d ( v ) ? d ( w ) where u, v, w are 01-words that encode the boundary conditions of ordinary TFPLs and d ( u ) is the number of cells in the Ferrers diagram associated with u. In the third part we consider TFPLs with d ( w ) - d ( u ) - d ( v ) = 0 , 1 ; in the first case their numbers are given by Littlewood-Richardson coefficients, but also in the second case we provide formulas that are in terms of Littlewood-Richardson coefficients. The proofs of these formulas are of a purely combinatorial nature.

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980s that the total number of (2n+1)x(2n+1) DASASMs is \prod_{i=0}^n (3i)!/(n+i)!, and a conjecture of Stroganov from 2008 that the ratio between the numbers of (2n+1)x(2n+1) DASASMs with central entry -1 and 1 is n/(n+1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980s, that for odd-order DASASMs is the last to have been proved.

A triangular gap of side 2 in a sea of dimers in a 60° angle

We consider a triangular gap of side 2 in a 60° angle on the triangular lattice whose sides are zigzag lines. We study the interaction of the gap with the corner as the rest of the angle being completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its five images in the sides of the angle. This provides a new aspect of the parallel between the correlation of gaps in dimer packings and electrostatics developed by the first author in previous work.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wus 80th birthday.

Linear relations of refined enumerations of alternating sign matrices

In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition some left and right columns are fixed. The main result is a simple linear relation between the number of nxn alternating sign matrices where the top row as well as the left and the right column is fixed and the number of nxn alternating sign matrices where the two top rows and the bottom row are fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only some top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely.

Another refinement of the Bender-Knuth (ex-)conjecture

We compute the generating function of column-strict plane partitions with parts in {1, 2,..., n}, at most c columns, p rows of odd length and k parts equal to n. This refines both Krattenthalers [C. Krattenthaler, The major counting of nonintersecting lattice paths and generating functions for tableaux, Mem. Amer. Math. Soc. 115 (552) (1995) vi+109] and the authors [I. Fischer, A method for proving polynomial enumeration formulas, J. Combin. Theory Ser. A (in press). Preprint, math.CO/0301103] refinement of the Bender-Knuth (ex-)conjecture. The result is proved by an extension of the method for proving polynomial enumeration formulas which was introduced by the author in I. Fischer [A method for proving polynomial enumeration formulas, J. Combin. Theory Ser. A (in press). Preprint, math.CO/0301103] to q-quasi-polynomials.