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$$$$$$$$$Abstract: 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.
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