2 problems

Let $C_n$ be the directed cycle on $n$ vertices with no sinks. Let $\mathbf{c}_0$ be a binary configuration with $|\mathbf{c}_0|>1$, and let $1\leq k\leq n$. For $1\leq i\leq n$, d…

Let $P_n$ be the directed path on $n$ vertices, let $\beta_i$ denote the $i$-th Betti number of its avalanche complex, and let a binary configuration be an initial configuration wh…