Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings
Abstract
We study diagonal billiard trajectories inside the Young diagram of an integer partition λ. A trajectory has slope ± 1, passes straight through sides shared by adjacent cells, and reflects from the exterior boundary until it closes. Let σ(λ) be the number of closed orbits. This extends the mirror-curve model of Chokwe sona sand drawings studied by Gerdes from rectangular grids to arbitrary Young diagrams. Let Gλ be the cell-adjacency graph of λ, with its natural planar embedding. We identify the billiard orbits with the components of the medial link of Gλ, and deduce that σ(λ) = 1 + dim ℬ(Gλ) = nullity L(Gλ) over 𝔽2, where ℬ is the binary bicycle space and L the mod-2 Laplacian. Writing λ□ for the diagram obtained by deleting the first row and column of λ, we further prove σ(λ) = 1 + nullity𝔽2 A(Gλ□). For rectangles this recovers Gerdes' formula σ(nm) = gcd(m,n) via identities for Fibonacci polynomials over 𝔽2, and in general it gives the characterization: σ(λ) = 1 if and only if λ□ has an odd number of domino tilings. Using the checkerboard bipartition of λ□, we decompose $σ(λ) - 1$ into a color-imbalance term, related to the BG-rank of Berkovich-Garvan, and an even rank-deficiency term. This yields parity restrictions and lower bounds for the orbit number, and shows that for any fixed d, asymptotically all partitions have more than d orbits. We also prove that σ(λ) is at most the Durfee length of λ, determine σ(n, n−1, …, 1) = ⌈ n/2 ⌉ for staircase partitions, and show that the adjacency-nullity formula is independent of the ground field.
Related
- Self-intersection Points of Billiard Trajectories in a Square with Small Pockets
- Trace formula for billiard flow on surfaces with concave boundary
- The Integrability of a knife-edge Billiard in a Disk
- Spectral Theory of Semisimple Bivariate Bicycle Codes
- Existence and Characterization of Bivariate Bicycle Codes
BibTeX
@misc{hemmer2026billiard,
title = {{Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings}},
author = {David J. Hemmer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16533},
eprint = {2609.16533},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16533},
}
RIS
TY - UNPB
TI - Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings
AU - David J. Hemmer
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16533
UR - https://arxiv.org/abs/2609.16533
AB - We study diagonal billiard trajectories inside the Young diagram of an integer partition λ. A trajectory has slope ± 1, passes straight through sides shared by adjacent cells, and reflects from the exterior boundary until it closes. Let σ(λ) be the number of closed orbits. This extends the mirror-curve model of Chokwe sona sand drawings studied by Gerdes from rectangular grids to arbitrary Young diagrams. Let G_λ be the cell-adjacency graph of λ, with its natural planar embedding. We identify the billiard orbits with the components of the medial link of G_λ, and deduce that σ(λ) = 1 + dim ℬ(G_λ) = nullity L(G_λ) over 𝔽_2, where ℬ is the binary bicycle space and L the mod-2 Laplacian. Writing λ^□ for the diagram obtained by deleting the first row and column of λ, we further prove σ(λ) = 1 + nullity_𝔽_2 A(G_λ^□). For rectangles this recovers Gerdes' formula σ(n^m) = gcd(m,n) via identities for Fibonacci polynomials over 𝔽_2, and in general it gives the characterization: σ(λ) = 1 if and only if λ^□ has an odd number of domino tilings. Using the checkerboard bipartition of λ^□, we decompose $σ(λ) - 1$ into a color-imbalance term, related to the BG-rank of Berkovich-Garvan, and an even rank-deficiency term. This yields parity restrictions and lower bounds for the orbit number, and shows that for any fixed d, asymptotically all partitions have more than d orbits. We also prove that σ(λ) is at most the Durfee length of λ, determine σ(n, n−1, …, 1) = ⌈ n/2 ⌉ for staircase partitions, and show that the adjacency-nullity formula is independent of the ground field.
ER -