3 problems

Let $P$ be a partition, let $\mathcal D_P$ be the associated poset, and let $C_a$ be a maximum-length U-chain in $\mathcal D_P$. Removing $C_a$ gives a partition $P'(P,a)$; recursi…

Let $K$ be a field, let $g$ be a nonzero polynomial in the noncommuting indeterminates $u_1,\ldots,u_{k+1}$ such that … for every $i\leq k$, the indeterminate $u_i$ appears exactly…

Let $A$ be a cyclic group of order $m$. For $\boldsymbol{\pi}=(\pi_1,\ldots,\pi_m)\in(\mathbf{Z}/m\mathbf{Z})^m$ with $\pi_1\ge\cdots\ge\pi_m$, an element $x\in A$ is…