The counts of parity-odd vs even late-growing permutations in the group 𝕊_2p−1 differ by (−1)^p

Abstract

A permutation σ∈𝕊_n of objects {1,…,n} is called \emph{late-growing} if its partial sums never exceed the respective partial sums of the `average' permutation (with the arithmetic mean n⁻¹·∑_i=1ⁿ i = (n+1)/2 of 1,…,n on average at each entry σ(i)): ∑_i=1^k σ(i) leqslant k· (n+1)/2 for all positive kleqslant n. We prove that for n=2p−1 odd, the set Φ_p⊆ 𝕊_2p−1 of late-growing permutations is partitioned into parity-even vs odd elements, Φ_p = Φ_p⁺ sqcup Φ_p⁻, in an almost symmetric way: | Φ_p⁺ | − | Φ_p⁻ | = (−1)^p+1. The proof is constructive: by viewing the permutations σ∈Φ_p as constrained walks on the integral lattice ℤ, we describe two filters that pair all but one element in Φ_p⁺ versus Φ_p⁻; the remaining walk σ^* with monotonically growing step amplitudes determines the dominant parity subset.

New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations
Kian C. Shah, Arthemy V. Kiselev
arXiv6 Oct 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity
Ryan J. Amaral, Juan B. Gil, Michael D. Weiner
arXiv28 Sep 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations
Lingsen Meng
arXiv9 Sep 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure
Tatsuya Goto
arXiv7 Oct 2026PreprintarXiv licenceFull textLogic & foundationsarXiv
A Short Proof of Zhang's Conjecture on Stack-Sortable Permutations
Maximus Lu
arXiv23 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{shah2026counts,
  title = {{The counts of parity-odd vs even late-growing permutations in the group $\mathbbS_2p-1$ differ by $(-1)^p$}},
  author = {Kian C. Shah and Arthemy V. Kiselev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35779},
  eprint = {2610.35779},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35779},
}

RIS

TY  - UNPB
TI  - The counts of parity-odd vs even late-growing permutations in the group 𝕊_2p−1 differ by (−1)^p
AU  - Kian C. Shah
AU  - Arthemy V. Kiselev
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35779
UR  - https://arxiv.org/abs/2610.35779
AB  - A permutation σ∈𝕊_n of objects {1,…,n} is called \emph{late-growing} if its partial sums never exceed the respective partial sums of the `average' permutation (with the arithmetic mean n^−1·∑_i=1^n i = (n+1)/2 of 1,…,n on average at each entry σ(i)): ∑_i=1^k σ(i) leqslant k· (n+1)/2 for all positive kleqslant n. We prove that for n=2p−1 odd, the set Φ_p⊆ 𝕊_2p−1 of late-growing permutations is partitioned into parity-even vs odd elements, Φ_p = Φ_p^+ sqcup Φ_p^−, in an almost symmetric way: | Φ_p^+ | − | Φ_p^− | = (−1)^p+1. The proof is constructive: by viewing the permutations σ∈Φ_p as constrained walks on the integral lattice ℤ, we describe two filters that pair all but one element in Φ_p^+ versus Φ_p^−; the remaining walk σ^* with monotonically growing step amplitudes determines the dominant parity subset.
ER  -

Abstract

Related

Cite