Triangulations of Δn−1×Δd−1\Delta_{n-1} \times \Delta_{d-1} and Tropical Oriented Matroids

Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.Comment: 11 pages and 3 figures. Any comment or feedback would be welcomed v2. Our result is that triangulations of product of simplices is a tropical oriented matroid. We are trying to extend this to all subdivisions. v3 Replaces the proof of Lemma 2.6 with a reference.. Proof of the matrix being totally unimodular is now more detailed. Extended abstract will be submitted to FPSAC '1

The Selberg integral and Young books

The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books" are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, product formulas for the number of standard Young tableaux of these shapes are obtained.Comment: 13 pages, 11 figure