2 problems

Let $A_i$ denote an antichain of size $i$, and let $P$ be a poset whose Hasse diagram is contained in the Hasse diagram of $A_k\oplus A_2$. Let $M^{P}$ be the promotion matrix of…

Let $P$ be a poset of size $n$ that is not a down forest, and let $M^{P}$ be its promotion transition matrix. For each element $x$ of $P$, call an element $y$ a successor of $x$ wh…