Discrete CATS Seminar
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**
UNIVERSITY OF KENTUCKY **

DISCRETE CATS SEMINAR

(where CATS = COMBINATORICS, ALGEBRA, TOPOLOGY AND STATISTICS!)

112 PATTERSON OFFICE TOWER

SPRING/SUMMER 2008

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DOCTORAL DEFENSE

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

Carlos Nicolas

University of Kentucky

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Thursday, July 3, 2008

10:00 am, 845 Patterson Office Tower

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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.