# Oddly beautiful plot $x=\binom{x}{y}$

I recently stumbled across the plot, which is interesting both close up and further out!

$$x=\binom{x}{y}$$

It produces this contour plot (via Wolfram Alpha):

plot (x = (x choose y)), x from -5 to 5, y from -5 to 5


plot (x = (x choose y)), x from -10 to 10, y from -10 to 10


plot (x = (x choose y)), x from -1000 to 1000, y from -1000 to 1000


Does this plot have a name? Why does it look this way? As well, are there any other plots like this, with this type of interesting behavior?

• @Moo Yep, an alternative command to plot them would be ContourPlot[{x == Binomial[x, y]}, {x, -5, 5}, {y, -5, 5}] (changing -5 and 5 to whatever plot dimensions). – esote Mar 9 '17 at 0:48
• WA is computing binomial coefficients using the gamma function to interpolate factorials, which has lots of strange behavior for negative values of its argument. – Qiaochu Yuan Mar 9 '17 at 0:57
• as Qiaochu said, wolfram is using the Gamma Function to plot this, which is why you see the strange behavior for negative x's. $x=\binom{x}{y} \,\Rightarrow x = \frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$ – Dando18 Mar 9 '17 at 1:05
• @QiaochuYuan Wow, you're right: imgur.com/Xbc7eMK, which is from Plot3D[Gamma[x]/Gamma[y], {x, -5, 0}, {y, -5, 0}] – esote Mar 9 '17 at 1:05

These plots are not quite correct. What's going on here is that ${x \choose y}$ is interpreted in terms of the Gamma function as $\frac{\Gamma(x+1)}{\Gamma(y+1) \Gamma(x-y+1)}$. I'll call this $c(x,y)$. However, $\Gamma$ has a singularity at nonpositive integers. The vertical lines at negative integer values of $x$ are due to those singularities. If $y$ is not an integer, $c(x,y) \to +\infty$ as $x$ approaches a negative integer value from one side and $+\infty$ as it approaches from the other side. The plotting software interprets that as a point on the curve.
Here's a somewhat more accurate plot produced by Maple. The green lines correspond to the singularities at negative integer values of $x$.