4 research outputs found
On an algebraicity theorem of Kontsevich
We give in a particular case a combinatorial proof of a recent algebraicity result of Kontsevich; the proof uses generalized one-sided and two-sided Dyck words, or equivalently, excursions and bridges. We indicate a noncommutative version of these notions, which could lead to a full proof. We show also a relation with pointed planar maps
Grand zigzag knight's paths
We study the enumeration of different classes of grand knight's paths in the
plane. In particular, we focus on the subsets of zigzag knight's paths subject
to constraints. These constraints include ending at ordinate 0, bounded by a
horizontal line, confined within a tube, among other considerations. We present
our results using generating functions or direct closed-form expressions. We
derive asymptotic results, finding approximations for quantities such as the
probability that a zigzag knight's path stays in some area of the plane, or for
the average of the final height of such a path. Additionally, we exhibit some
bijections between grand zigzag knight's paths and some pairs of compositions.Comment: 21 pages, 9 figure