3 problems

Let $p$ be a prime, let $n$, $j$, and $k$ be positive integers with $j<n$, and let $F\in M_k(\Gamma^{(n)}_{1}(N))_{\mathbb{Z}_{(p)}}$ with $k\in\mathbb{Z}_{\ge 1}$. Assume that…

Let $n$ be a positive integer, let $\mathsf{RTT}j(1^{n+1})$ denote the set of standard fully tiered rooted trees on $n+1$ vertices whose root is labelled $j$, and let…

Let $u$ be a composition, let $u( u)$ denote the partition obtained by rearranging its parts, and let $\mathsf{RTT}(nu)$ be the set of fully tiered rooted trees associated with…