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A Note on Log-Convexity of q-Catalan Numbers
Lynne M. Butler1 and W. Patrick Flanigan2
1Department of Mathematics, Haverford College, Haverford, PA 19041-1392, USA
2Blazek & Gebhardt Retirement Services, Inc., Suite 201, Timonium, MD, 21093, USA
Annals of Combinatorics 11 (3-4) p.369-373 September, 2007
AMS Subject Classification:05A20, 05A10, 05A05, 11B65
The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefcients. They evaluate, at q = 1, to the Catalan numbers: 1, 1, 2, 5, 14, …, a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted.
Keywords: Catalan, log-convex, Gaussian, log-concave, q-log-convex, q-log-concave


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