Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings

Source abstract

We study diagonal billiard trajectories inside the Young diagram of an integer partition $λ$. A trajectory has slope $\pm 1$, 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 $σ(λ) = 1 + \dim \mathcal{B}(G_λ) = \mathrm{nullity}\, L(G_λ)$ over $\mathbb{F}_2$, where $\mathcal{B}$ is the binary bicycle space and $L$ the mod-2 Laplacian. Writing $λ^\square$ for the diagram obtained by deleting the first row and column of $λ$, we further prove $σ(λ) = 1 + \mathrm{nullity}_{\mathbb{F}_2} A(G_{λ^\square})$. For rectangles this recovers Gerdes' formula $σ(n^m) = \gcd(m,n)$ via identities for Fibonacci polynomials over $\mathbb{F}_2$, and in general it gives the characterization: $σ(λ) = 1$ if and only if $λ^\square$ has an odd number of domino tilings. Using the checkerboard bipartition of $λ^\square$, 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 $σ(n, n-1, \ldots, 1) = \lceil n/2 \rceil$ for staircase partitions, and show that the adjacency-nullity formula is independent of the ground field.