We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a by-product of investigations on various combinatorial objects. For exa...

A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and co...

Digital halftoning is a technique to convert a continuous-tone image into a binary image consisting of black and white dots. It is an important technique for printing machines and printers to output an image with few intensity levels or colors which looks similar to an input image. In this talk I will explain how computational geometry and combinatorial optimization can contribute to digital ha...

Authors Yu. I. Manin and V. V. Schechtman developed the theory of “higher Bruhat orders,” presenting a family of combinatorial objects closely related to the symmetric group. In particular, the two authors define a series of ranked posets which generalize the weak left Bruhat order; their construction has found applications in geometry and representation theory. In this paper, we define a simil...

The combinatorial q, t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q, t-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n’th q, t-Catalan number is the Hilb...

The combinatorial q, t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q, t-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n’th q, t-Catalan number is the Hilb...

The combinatorial q, t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q, t-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n’th q, t-Catalan number is the Hilb...

P aul Erdős died September 20, 1996, and a memorial article appears elsewhere in this issue. This feature article gives a cross section of his monumental oeuvre. Most of Erdős’s work falls roughly into the following categories: • number theory • finite combinatorics (including graph theory) • combinatorial geometry • set theory, set-theoretical topology • constructive theory of functions (appro...

We describe a combinatorial virtual screening approach for predicting high specificity heparin/heparan sulfate sequences using the well-studied antithrombin-heparin interaction as a test case. Heparan sulfate hexasaccharides were simulated in the 'average backbone' conformation, wherein the inter-glycosidic bond angles were held constant at the mean of the known solution values, irrespective of...

We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part of classical Fourier analysis that investigates the relationship between geometric and Fourier-analytic pro...