Year: 2026
Type: preprint
Abstract: Place one grain at every nonsink vertex of the wired $n\times n$ 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 $n\ge1$, \[ ν_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ ν_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] 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 $\mathbb F_2$, giving the exact valuation.
Source: arXiv (Cornell University)
Institutions: Middle East Technical University, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Technion – Israel Institute of Technology, Guangdong Technion-Israel Institute of Technology, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Moscow Institute of Physics and Technology
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Science Foundation Ireland
OpenAlex ID: W7212322713