## Enumerations of permutations by circular descent sets.(English)Zbl 1445.05009

Summary: The circular descent of a permutation $$\sigma$$ is a set $$\{ \sigma(i) \mid \sigma(i) > \sigma(i+1) \}$$. In this paper, we focus on the enumerations of permutations by the circular descent set. Let $$\operatorname{cdes}_n(S)$$ be the number of permutations of length $$n$$ which have the circular descent set $$S$$. We derive the explicit formula for $$\operatorname{cdes}_n(S)$$. We describe a class of generating binary trees $$T_k$$ with weights. We find that the number of permutations in the set $$\operatorname{CDES}_n(S)$$ corresponds to the weights of $$T_k$$. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.

### MSC:

 05A15 Exact enumeration problems, generating functions 05A05 Permutations, words, matrices 05E10 Combinatorial aspects of representation theory
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### References:

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