Shader, Sarah (School: DaVinci Academy of the Science and the Arts)
The weighted Catalan numbers, like the Catalan numbers, enumerate various mathematical objects. For example, the number of Morse links with n critical points is the n-th weighted Catalan number, L_n, with weights 1^2, 3^2, 5^2,...,(2k+1)^2,.... This paper examines the conjecture made by Postnikov which involves examining the divisibility of L_n by powers of 3. This project gives an upper bound of 2*3^(2r-7) on the period of L_n modulo 3^r, which supports Postnikov's conjecture that this period is 2*3^(r-3). The results are proven by representing L_n using combinatorial structures called Dyck paths. Dyck paths of length n are broken into pieces using a process called partial flat path decomposition. This classifies paths according to the location of the steps corresponding to the weights divisible by 3^2 or the weight 1. Properties of partial flat paths are proven and this knowledge combined with the use of mathematical tools, specifically generating functions, lead to the main result.
American Mathematical Society: Second Award of $1000