Why does the beta function look like a Sudoku grid?

Question

I understand the beta function is defined by:

$B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$

So in my mind it is kind of an inverse continuous choice function. Since

$ ^{x+y}C_r = {n \choose x} = {(x+y)! \over x!y!} $

What intuition sheds light on why the real plot of it looks like a Sudoku grid?

Answer 1

I suppose the "grid" you mean refers to the horizontal and vertical lines or gaps in the contour plot of the beta function as seen here or here.

The reason for these lines is that the gamma function goes to infinity at zero and at every negative integer. This leads to the beta function going to infinity at most points along the lines $x=k$ and $y=k$ for $k$ any non-positive integer. For example, in neighborhoods of most points on the line $x=-1$ there are finite values of $\Gamma(y)$ and finite non-zero values of $\Gamma(x+y),$ hence as we approach the point on the line the beta function goes to infinity.