A Note on Log-Convexity of q-Catalan Numbers

Lynne M. Butler^{1} and W. Patrick Flanigan^{2}

lbutler@haverford.edu

wflaniga@gmail.com

Annals of Combinatorics 11 (3-4) p.369-373 September, 2007

Abstract:

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.
References:

1. L.M. Butler, The q-log-concavity of q-binomial coefficients, J. Combin. Theory Ser. A 54 (1) (1990) 54-63.

2. L. Carlitz and J. Riordan, Two element lattice permutations and their q-generalization, Duke J. Math. 31 (1964) 371-388.

3. J. Fürlinger and J. Hofbauer, q-Catalan numbers, J. Combin. Theory Ser. A 40 (2) (1985) 248-264.

4. L. Liu and Y. Wang, On the log-convexity of combinatorial sequences, Adv. Appl. Math., to appear.

5. P.A. MacMahon, Combinatory Analysis, Vol. I, Cambridge University Press, Cambridge, 1915.