  Let $p$ be a prime. We study the structure of and the inclusion relations
among the terms in the monomial lattice in the modular symmetric power
representations of $\mathrm{GL}_2(\mathbb{F}_p)$. We also determine the
structure of certain related quotients of the symmetric power representations
which arise when studying the reductions of local Galois representations of
slope at most $p$. In particular, we show that these quotients are periodic and
depend only on the congruence class modulo $p(p-1)$. Many of our results are
stated in terms of the sizes of various sums of digits in base $p$-expansions
and in terms of the vanishing or non-vanishing of certain binomial coefficients
modulo $p$.
