DOCTORAL DEFENSE
"Structural and enumerative properties of k-triangulations of the n-gon."

Carlos Nicolas
University of Kentucky

Thursday, July 3, 2008
10:00 am, 845 Patterson Office Tower

Abstract:

A k-triangulation of the n-gon is a maximal set of diagonals of the n-gon such that no k+1 of them cross each other. The case k=1 coincides with the usual triangulations of the n-gon, and some of their properties have been known to hold for the general case. In this talk, I will recount these properties and prove a few more that I found. In 2004, Jonsson showed that the number of k-triangulations of the n-gon is given by a Hankel determinant of Catalan numbers. This determinant also counts the number of k non-crossing Dyck paths of size n-2k. A combinatorial bijection between these two sets is known only for the cases k=1 and k=2. I will construct a bijection for the case k=2 that preserves two pairs of parameters which are natural to these objects. As a consequence I obtain a refinement, which involves Catalan and ballot numbers, of the formula for the number of 2-triangulations. I will make some conjectures for the general case.

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The Discrete CATS seminar is supported in part by the Department of Mathematics and the College of Arts and Sciences through the College Enrichment Fund. We also would like to thank Richard Ehrenborg, who generously supports speakers with his NSA grant.

If you wish to be added to the seminar list, please e-mail Margaret Readdy at readdy (at) ms.uky.edu