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It has been suggested to me that I use this notation: $$ \lfloor n \rfloor_2 = 2 \left\lfloor \frac n 2 \right\rfloor = \text{“even floor of $n$''} = \text{largest even integer}\le n. $$

I also want to write about an “even falling factorial” that, for example, given the inputs $57$ and $6$, or $56$ and $6$, has this value: $$ 56\times54\times52\times50\times48\times46, $$ i.e. it is $$ \lfloor 57 \rfloor_2 \cdot (\lfloor 57 \rfloor_2 - 2) \cdot (\lfloor 57 \rfloor_2 - 4) \cdot (\lfloor 57 \rfloor_2 - 6) \cdot (\lfloor 57 \rfloor_2 - 8) \cdot (\lfloor 57 \rfloor_2 - 10)$$ so in general, given $n$ and $k$, it is this: $$ \prod_{j=0}^{k-1} \lfloor n - 2j \rfloor_2. $$ I could just call it $n\mathbin{\sharp}k$ or something like that. But my questions are:

  • Is there some standard notation for this?; and
  • What notation would be easiest for the reader to follow when the topic is neither the notation nor the concept that it denotes but rather the notation and the concept are merely being used in the course of discussing a topic for which they are useful?
Michael Hardy
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    Mathworld has a notation for the generalized falling factorial at http://mathworld.wolfram.com/FallingFactorial.html But I can't say I'm a big fan of the notation there. – Mark S. Mar 07 '16 at 19:13

2 Answers2

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In analogy with the usual definitions of partial permutation numbers in terms of (normal) factorials, you could compactly express this quantity as $$ \frac{\lfloor n \rfloor_2!!}{\lfloor n - 2j \rfloor_2!!}, $$ where $!!$ denotes the double factorial.

vadim123
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Michael Seifert
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Could you just define it as a two-variable function $f(n,k)?$

You could avoid introducing new notation if you wrote it

$$f(n,k)=\displaystyle\prod_{i=0}^{k-1} \left(2\left\lfloor\frac{n}{2}\right\rfloor - 2i\right)$$

Another way would be

$$f(n,k) = k! \cdot 2^k \cdot \binom{\left\lfloor\frac{n}{2}\right\rfloor}{k}$$

Zubin Mukerjee
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