The circular descent of a permutation σ is a set {σ(i) | σ(i) > σ(i + 1)}. In this paper, we focus on the enumerations of permutations by the circular descent set. Let cdes n (S) be the number of permutations of length n which have the circular descent set S. We derive the explicit formula for cdes n (S). We describe a class of generating binary trees T k with weights. We find that the number of permutations in the set CDES n (S) corresponds to the weights of T k . As a application of the main

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... esults in this paper, we also give the enumeration of permutation tableaux according to their shape.