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Let $\mathcal S$ be a subset of the positive integers, and $M$ be a positiveinteger. Mohammad K. Azarian, inspired by work of Tony Colledge, considered thenumber of ways to climb a staircase containing $n$ stairs using "step-sizes" $s\in \mathcal S$ and multiplicities at most $M$.  In this exposition, we find a solution via generating functions, i.e., anexpression which counts the number of partitions $n = \sum_{s \in \mathcal S}m_s s$ satisfying $0 \leq m_s \leq M$. We then use this result to answer aseries of questions posed by Azarian, thereby showing a link with ten sequenceslisted in the On-Line Encyclopedia of Integer Sequences. We conclude by posingopen questions which seek to count the number of compositions of $n$.

On the Generalized Climbing Stairs Problem