# Understanding Binomial coefficient with floored terms

I was reading through the notation used in a paper on arxiv.org when I came across this on page 6:

$$[x]$$ the floor of $$x$$

$$\{x\}$$ the sawtooth function of $$x$$. That is $$\{x\} = x - [x]$$

$$\begin{Bmatrix}x\\y \end{Bmatrix}$$ Binomial coefficient with floored terms.

Here is the explanation:

That is, $$\begin{Bmatrix}x\\y\end{Bmatrix} = \delta(y,x){{[x]}\choose{[y]}}$$

where:

• $$\delta(y,x)=1$$ if $$\{x\} \ge \{y\}$$

• $$\delta(y,x)=[x-y]+1$$ if $$\{x\} < \{y\}$$

Does this definition make sense? If so, could someone help me to understand what it means when $$\delta(y,x) \neq 1$$?

• What part doesn’t make sense to you? For example, try computing $\genfrac{\{}{\}}{0pt}{}{7.4}{3.6}$ and tell us where you get stuck. – Anders Kaseorg Jan 8 at 9:19
• $\begin{Bmatrix}7.4\\ 3.5\end{Bmatrix} = (4){7\choose3}$. I am clear on the computation. I'm not clear why the $4$ is needed. Why not just $\begin{Bmatrix}7.4\\ 3.5\end{Bmatrix} = {7\choose3}$ – Larry Freeman Jan 8 at 11:40

The author is of course free to make any definition they want, and apparently they found $$\delta(y, x)\binom{\lfloor x\rfloor}{\lfloor y\rfloor}$$ to be useful for their purposes in a way that $$\binom{\lfloor x\rfloor}{\lfloor y\rfloor}$$ alone was not. See Lemma 2.0.2 for a hint of why that might be:
$$\genfrac{\{}{\}}{0em}{}{x}{y} = \frac{\prod_{k \in (s - r, s] \cap \mathbb N} k}{\prod_{k \in (0, r] \cap \mathbb N} k} = \delta(y, x)\binom{\lfloor x\rfloor}{\lfloor y\rfloor}.$$