完全二部并行芯片点火的中间楼梯
The Middle Stair for Complete Bipartite Parallel Chip-Firing
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中文总结 AI 辅助
该研究证明完全二部并行芯片点火的中间楼梯猜想，引入双参数共轭方法，得出特定构型下游戏最终周期为2的结论。
中文摘要 AI 辅助
我们证明了所有完全二部图的中间楼梯猜想。若完全二部图K_{a,b}上的并行芯片点火游戏具有构型σ，满足2ab-a-b<|σ|<2ab，则其最终周期为2。平衡情形K_{a,a}已由Ji、Li和Wang利用单参数共轭构型证明。我们引入双参数共轭c^{k,ℓ}，其中一侧的秩偏移为另一侧提供加性偏移，这些共轭保持芯片总数和活性。精确的Ferrers图计数在一侧产生具有两轮点火覆盖的非负共轭，该覆盖以交替两轮波传播，活性为1/2；非团性则强制周期为2。
英文摘要
We prove the middle-stair conjecture for every complete bipartite graph. If a parallel chip-firing game on $K_{a,b}$ has configuration $σ$ with $2ab-a-b<|σ|<2ab$, then its eventual period is $2$. The balanced case $K_{a,a}$ was proved by Ji, Li, and Wang using one-parameter conjugate configurations. We introduce two-parameter conjugates $c^{k,\ell}$, in which the rank shift on one side supplies the additive offset on the other. These conjugates preserve both the total number of chips and the activity. An exact Ferrers-diagram count then produces a nonnegative conjugate with two-round firing coverage on one side. The coverage propagates in alternating two-round waves, giving activity $1/2$; non-clumpiness then forces period $2$.
