Inflection sets of permutations
Abstract
In 2020, Brualdi and Dahl introduced the discrete derivative σ'(i)=σ(i+1)−σ(i) of a permutation σ∈Symn. In this paper, we focus on the second derivative of a permutation, σ''(i)=σ(i+1)+σ(i−1)−2σ(i), and the inflections it determines. We show that for even n, the only distinct permutations with the same second derivative are the identity and its reverse, while for odd n, the second derivative is not injective even after excluding them. We then study which sets of pairs of integers (i,j) are the inflection set of some permutation and whether a given set is witnessed in Symn for infinitely many n. We show that exactly $14$ permutations of [n] have no inflections when n≥5. For a single inflection {(k,m)}, we determine all n for which it is witnessed in Symn: we prove it is always witnessed, and it is witnessed for only finitely many n if and only if m≥ 2k+1. In that case, all such n satisfy 2m−2k+1≤ n≤ 2m−k+2 (apart from k=2, n=m+1), and, perhaps surprisingly, this range has gaps once k≥ 10, the smallest occurring at (k,m,n)=(10,21,30). Finally, we characterize maximally inflected permutations, whose second derivative alternates in sign, and show that their number |ℳn| satisfies 2En≤|ℳn|≤(2/3)⌊ n/3⌋−1 n!, where En is the Euler zigzag number.
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BibTeX
@misc{kirby2026inflection,
title = {{Inflection sets of permutations}},
author = {Gordon Rojas Kirby and Mariana Smit Vega Garcia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38237},
eprint = {2610.38237},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38237},
}
RIS
TY - UNPB
TI - Inflection sets of permutations
AU - Gordon Rojas Kirby
AU - Mariana Smit Vega Garcia
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38237
UR - https://arxiv.org/abs/2610.38237
AB - In 2020, Brualdi and Dahl introduced the discrete derivative σ'(i)=σ(i+1)−σ(i) of a permutation σ∈Sym_n. In this paper, we focus on the second derivative of a permutation, σ''(i)=σ(i+1)+σ(i−1)−2σ(i), and the inflections it determines. We show that for even n, the only distinct permutations with the same second derivative are the identity and its reverse, while for odd n, the second derivative is not injective even after excluding them. We then study which sets of pairs of integers (i,j) are the inflection set of some permutation and whether a given set is witnessed in Sym_n for infinitely many n. We show that exactly $14$ permutations of [n] have no inflections when n≥5. For a single inflection {(k,m)}, we determine all n for which it is witnessed in Sym_n: we prove it is always witnessed, and it is witnessed for only finitely many n if and only if m≥ 2k+1. In that case, all such n satisfy 2m−2k+1≤ n≤ 2m−k+2 (apart from k=2, n=m+1), and, perhaps surprisingly, this range has gaps once k≥ 10, the smallest occurring at (k,m,n)=(10,21,30). Finally, we characterize maximally inflected permutations, whose second derivative alternates in sign, and show that their number |ℳ_n| satisfies 2E_n≤|ℳ_n|≤(2/3)^⌊ n/3⌋−1 n!, where E_n is the Euler zigzag number.
ER -