Enumerations of Permutations by Circular Descent Sets

TL;DR

This paper derives explicit formulas for counting permutations with specific circular descent sets and connects these counts to weighted binary trees, also applying results to permutation tableaux enumeration.

Contribution

It introduces a new explicit formula for enumerating permutations by circular descent sets and links these counts to weighted binary trees and permutation tableaux.

Findings

Derived explicit formulas for $cdes_n(S)$

Connected permutation counts to weighted binary trees

Enumerated permutation tableaux by shape

Abstract

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 $cdes_n(S)$ 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 results in this paper, we also give the enumeration of permutation tableaux according to their shape.