Taiwanese Journal of Mathematics

Enumerations of Permutations by Circular Descent Sets

Hungyung Chang, Jun Ma, and Jean Yeh

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The circular descent of a permutation $\sigma$ is a set $\{ \sigma(i) \mid \sigma(i) \gt \sigma(i+1) \}$. 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.

Article information

Taiwanese J. Math., Advance publication (2019), 13 pages.

First available in Project Euclid: 28 January 2019

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Digital Object Identifier

Primary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

circular descent generating tree permutation permutation tableaux


Chang, Hungyung; Ma, Jun; Yeh, Jean. Enumerations of Permutations by Circular Descent Sets. Taiwanese J. Math., advance publication, 28 January 2019. doi:10.11650/tjm/190105. https://projecteuclid.org/euclid.twjm/1548666174

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