# Descents in $t$-Sorted Permutations

## TL;DR

This paper investigates the maximum number of descents in t-sorted permutations generated by West's stack-sorting map, providing exact bounds, characterizations, and enumeration for permutations attaining these bounds.

## Contribution

It establishes the maximum descents in t-sorted permutations, characterizes those reaching this maximum, and counts them explicitly.

## Key findings

- Maximum descents in t-sorted permutations is ⌊(n−t)/2⌋.
- Characterization of t-sorted permutations attaining maximum descents when n and t have same parity.
- Number of such permutations is (n−t−1)!!.

## Abstract

Let $s$ denote West's stack-sorting map. A permutation is called $t-\textit{sorted}$ if it is of the form $s^t(\mu)$ for some permutation $\mu$. We prove that the maximum number of descents that a $t$-sorted permutation of length $n$ can have is $\left\lfloor\frac{n-t}{2}\right\rfloor$. When $n$ and $t$ have the same parity and $t\geq 2$, we give a simple characterization of those $t$-sorted permutations in $S_n$ that attain this maximum. In particular, the number of such permutations is $(n-t-1)!!$.