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 $\delta(y,x) \neq 1$?
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:
$$\genfrac{\{}{\}}{0em}{}{x}{y} = \frac{\prod_{k \in (s - r, s] \cap \mathbb N} k}{\prod_{k \in (0, r] \cap \mathbb N} k} = \delta(y, x)\binom{\lfloor x\rfloor}{\lfloor y\rfloor}.$$