Modelling x-ray tomography using integer compositions

Abstract: The x-ray process is modelled using integer compositions as a two dimensional analogue of the object being x-rayed, where the examining rays are modelled by diagonal lines with equation $x-y=n$ for non negative integers $n$. This process is essentially parameterised by the degree to which the x-rays are contained inside a particular composition. So, characterising the process translates naturally to obtaining a generating function which tracks the number of "staircases" which are contained inside arbitrary integer compositions of $n$. More precisely, we obtain a generating function which counts the number of times the staircase $1^+2^+3^+\cdots m^+$ fits inside a particular composition. The main theorem establishes this generating function \begin{equation*} F= \dfrac {k_{m}-\frac {qx^{m}y}{1-x}k_{m-1}}{(1-q)x^{\binom {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.