Perfect powers in Catalan and Narayana numbers

Abstract

When a Catalan number or a Narayana number is a (non-trivial) perfect power? For Catalan numbers, we show that the answer is "never". However, we prove that for every b, the Narayana number N(a,b) is a (non-trivial) perfect square for infinitely many values of a, and we show how to compute all of them. We also conjecture that N(a,b) is never a (non-trivial) perfect k-th power for k greater than 2 and we prove some cases of this conjecture.

15 pages, 2 figures

References in corpus (5)

• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers
• Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial
• Generalized Chung-Feller Theorems for Lattice Paths (Thesis)
• Combinatorics of Labelled Parallelogram polyominoes