In the same idea as of this question on factorials; since we showed we can build linear systems for calculating factorials, maybe in the same way we can build linear equation systems to verify theorems in combinatorics?
Own work so far:
We can for example define the Pascal's triangle recursively on a grid:
$$\text{Boundary condition: }\hspace{0.125cm}g = \left[\begin{array}{c|c|c|c|c}1&1&1&1&1\\\hline1&0&0&0&0\\\hline1&0&0&0&0\\\hline1&0&0&0&0\\\hline1&0&0&0&0\end{array}\right]\hspace{0.125cm} \text{ and filter: } f = \left[\begin{array}{r|r}-1&1\\\hline1&0\end{array}\right]$$ we can set up the least squares problem: $${\bf v_o} =\min_{\bf v}\{\|{\bf Fv}\|_2^2 + {\bf C_g}\|{\bf v-g}\|_2^2\}$$ Where $\bf F$ is the matrix representation of the convolution operator of filter $f$. $\bf g$ is a vectorization of the boundary condition values and $\bf C_g$ is a diagonal matrix encoding which grid points are on the boundary.
The solution will propagate "automatically" giving us the well known binomial coefficients. In the same idea we can get multinomial coefficients, but we need to handle data and filters in more dimensions.
Is this enough or do we need to step outside of our comfy linear framework to prove anything of interest?