We show that the formalism of overpartitions gives a simple involution for the product definition of the Gaussian coefficients. In the formulation of our involution, an overline in the representation of an overpartition is endowed with a weight.

We prove the log-concavity of the Fennessey-Larcombe-French sequence based on its three-term recurrence relation, which was recently conjectured by Zhao. The key ingredient of our approach is a sufficient condition for log-concavity of a sequence subject to certain three-term recurrence.

Based on the combinatorial proof of Schur's partition theorem given by Bressoud, and the combinatorial proof of Alladi's partition theorem given by Padmavathamma, Raghavendra and Chandrashekara, we obtain a bijective proof of a partition theorem of Alladi, Andrews and Gordon.

Let {P n } n≥0 denote the Catalan-Larcombe-French sequence, which naturally came from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence { n √ P n } n≥1 , which was originally conjectured by Z. W. Sun. We also obtain the strict log-concavity of the sequence { n √ V n } n≥1 , where {V n } n≥0 is the Fennessey-LarcombeFrench sequence arising from the series expansion of the complete elliptic integral of the second kind.