  The aim of this paper is twofold. First, we study the number of partitions of
a positive integer $m$ into at most $n$ parts in a given set $A$. We prove that
such a number is bounded by the $n$-th Fibonacci number $F(n)$ for any $m$ and
some family of sets $A$ including sets of powers of an integer. Then, in the
second part of the paper, we provide new results in bounding the cohomology of
the simple algebraic group $SL_2$ with coefficients in Weyl modules.
