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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 $\delta(y,x) \neq 1$?

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  • $\begingroup$ 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. $\endgroup$ – Anders Kaseorg Jan 8 at 9:19
  • $\begingroup$ $\begin{Bmatrix}7.4\\ 3.5\end{Bmatrix} = (4){7\choose3}$. I am clear on the computation. I'm not clear why the $4$ is needed. Why not just $\begin{Bmatrix}7.4\\ 3.5\end{Bmatrix} = {7\choose3}$ $\endgroup$ – Larry Freeman Jan 8 at 11:40
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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:

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

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  • $\begingroup$ Thanks. I was trying to understand if this was a standard definition or a definition specific to this paper. $\endgroup$ – Larry Freeman Jan 8 at 12:51

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