In the present paper we study general properties of good sequences by means of a powerful and beautiful tool of combinatorics - the method of bijective proofs. A good sequence is a sequence of positive integers k=1,2,... such that the element k occurs before the last occurrence of k+1. We construct two bijections between the set of good sequences of fixed length and the set of permutations of the same length. This allows us to count good sequences as well as to calculate generating functions of statistics on good sequences. We study avoiding patterns on good sequences and discuss their relation with Eulerian polynomials. Finally, we describe particular interesting properties of permutations, again using bijections.

Author Bio

I am currently a student at the Lyc攼㸹e Louis-Le-Grand in Paris in Classes Préparatoires, which corresponds to the first year of University. The present paper summarizes a research project that I started at the Clay Research Academy 2005 under the supervision of Prof. Richard P. Stanley and Dr. Federico Ardila and completed after returning back from the Academy. At various stages of this work, I benefited from e-mail exchanges with Federico whom I met again at the International Mathematical Olympiad in July 2005. Back to France, numerous discussions with Xavier Caruso, a graduate student at the Ecole Normale Supérieure in Paris, contributed to improve the paper. In my spare time, I enjoy swimming and reading, especially science fiction.