arXiv: Combinatorics | 2019
Descents in $t$-Sorted Permutations.
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 $(n-t-1)!!$.