Peaks are preserved under run-sorting

TL;DR

This paper investigates run-sorting on permutations, revealing a bijection that preserves peak-values, analyzing the expected number of descents, and providing new formulas and properties for run-sorted permutations and binary words.

Contribution

It introduces a bijection preserving peak-values under run-sorting, derives the expected descents, and offers a closed-form generating function for run-sorted permutations.

Findings

Expected descents after run-sorting is (n-2)/3.

Descent polynomials are real rooted and interlaced.

Provides a new interpretation of OEIS sequence A124324.

Abstract

We study a sorting procedure (run-sorting) on permutations, where runs are rearranged in lexicographic order. We describe a rather surprising bijection on permutations on length $n$, with the property that it sends the set of peak-values to the set of peak-values after run-sorting. We also prove that the expected number of descents in a permutation $\sigma \in S_{n}$ after run-sorting is equal to $(n-2)/3$. Moreover, we provide a closed form of the exponential generating function introduced by Nabawanda, Rakotondrajao and Bamunoba in 2020, for the number of run-sorted permutations of $[n]$, ($RSP(n)$) having $k$ runs, which gives a new interpretation to the sequence A124324 in the Online Encyclopedia of Integer Sequences. We show that the descent generating polynomials, $A_{n}(t)$ for $RSP(n)$ are real rooted, and satisfy an interlacing property similar to that satisfied by the Eulerian polynomials. Finally, we study run-sorted binary words and compute the expected number of descents after run-sorting a binary word of length $n$.