The main theorem establishes the generating function
F which counts the number of times the staircase
fits inside an integer composition of
n.
\begin{equation*}
F= \dfrac {k_{m}-\frac {qx^{m}y}{1-x}k_{m-1}}{(1-q)x^{\binom
... [Show full abstract] {m+1}{2}}\left(\frac{y}{1-x}\right)^{m}+\frac{1-x-xy}{1-x}\left(k_{m}-\frac{qx^{m}y}{1-x}k_{m-1}\right)}.
\end{equation*}
where
\begin{equation*}
k_{m}=\sum_{\j=0}^{m-1}x^{mj-\binom {j}{2}}\left(\frac {y}{1-x}\right)^{j}.
\end{equation*}
Here x and y respectively track the composition size and number of parts, whilst q tracks the number of such staircases contained.