Peaks are preserved under run-sorting

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 (also known as the pinnacle set) to the set of peak-values after run-sorting. We also prove that the expected number of descents in a permutation σ ∈ Sn 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 http://oeis.org/A124324. We show that the descent generating polynomials, An(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.