| A sequence of real numbers ;We also give a clear exposition of a powerful involution due to Foata and Schutzenberger. This involution allows us to give a combinatorial explanation of the fact that two Ferrers boards are rook equivalent if and only if they are q-rook equivalent. We also show that for a fixed Ferrers board ;In the last part we introduce the concepts of log-Fibonacci and strong log-Fibonacci sequences. A sequence is log-Fibonacci if at the even indices we have log-concavity and at the odd indices we have log-convexity or vice-versa. We give a necessary and sufficient condition for an increasing sequence to be strongly log-Fibonacci. As examples of strongly log-Fibonacci sequences we have the Fibonacci numbers and a sequence of partitions.;In this work we generalize a method of Bender and Canfield for constructing new log-concave sequences from a given log-concave sequence. Our result applies to any sequence of objects which is strongly log-concave. Our proof, which uses symmetric functions, is different from the one presented by Bender and Canfield. |
