Taiwanese Journal of Mathematics

Enumerations of Permutations by Circular Descent Sets

Hungyung Chang, Jun Ma, and Jean Yeh

Full-text: Open access

Abstract

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

Source
Taiwanese J. Math., Volume 23, Number 6 (2019), 1303-1315.

Dates
Received: 19 November 2018
Revised: 10 January 2019
Accepted: 15 January 2019
First available in Project Euclid: 28 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1548666174

Digital Object Identifier
doi:10.11650/tjm/190105

Mathematical Reviews number (MathSciNet)
MR4033546

Zentralblatt MATH identifier
07142974

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

Keywords
circular descent generating tree permutation permutation tableaux

Citation

Chang, Hungyung; Ma, Jun; Yeh, Jean. Enumerations of Permutations by Circular Descent Sets. Taiwanese J. Math. 23 (2019), no. 6, 1303--1315. doi:10.11650/tjm/190105. https://projecteuclid.org/euclid.twjm/1548666174


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