The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square
Abstract
Place one grain at every nonsink vertex of the wired n× n square, and let L(n) be the order of this operation in the sandpile group. Thus L(n) is the least positive q for which q uniform grain layers form an integral combination of toppling moves. We prove that, for every n≥1, ν2(L(n))= 2,&n=1, 1,&n≥2 even, ν2(n+1)+2,&n≥3 odd. For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over 𝔽2, giving the exact valuation.
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BibTeX
@misc{akyar20262adic,
title = {{The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square}},
author = {Turgay Akyar and Artem Beliakov and Konstantin Delchev and Nikita Kalinin and Ernesto Lupercio and Higinio Serrano and Mikhail Shkolnikov and Daniel Tabares and Nikolai Terekhov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10625},
eprint = {2609.10625},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10625},
}
RIS
TY - UNPB
TI - The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square
AU - Turgay Akyar
AU - Artem Beliakov
AU - Konstantin Delchev
AU - Nikita Kalinin
AU - Ernesto Lupercio
AU - Higinio Serrano
AU - Mikhail Shkolnikov
AU - Daniel Tabares
AU - Nikolai Terekhov
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10625
UR - https://arxiv.org/abs/2609.10625
AB - Place one grain at every nonsink vertex of the wired n× n square, and let L(n) be the order of this operation in the sandpile group. Thus L(n) is the least positive q for which q uniform grain layers form an integral combination of toppling moves. We prove that, for every n≥1, ν_2(L(n))= 2,&n=1, 1,&n≥2 even, ν_2(n+1)+2,&n≥3 odd. For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over 𝔽_2, giving the exact valuation.
ER -