What are the bounds of a ratio with a given set of numbers and variables?

[iScience]

Sorry in advance if I've posted in the wrong section.

given the set $\{r_i, r_{ii}, r_{iii}, ... , r_R\}$
where $r \ \epsilon \ \mathbb{Z}_+ \ , \ r_i \geq r_{i+1}$How would you go about finding the bounds of something like this, or determining if it even has any?

$( \, log_2{\frac{(\sum_{i=1}^R{r_i})!}{\prod_{i=1}^R{(r_i!)}}}) \, :\sum_{i=1}^{R}{i \cdot r_i}$

 

[tommyxu3]

Since $r_i$ is the largest and $r_R$ is the smallest, the ratio in the log must smaller than $\frac{(r_iR)!}{(r_R!)^R},$ which may depend on $r_i$, $r_R$ and $R.$ Does the problem require a fixed constant?

 

[iScience]

Since $r_i$ is the largest and $r_R$ is the smallest, the ratio in the log must smaller than $\frac{(r_iR)!}{(r_R!)^R},$ which may depend on $r_i$, $r_R$ and $R.$ Does the problem require a fixed constant?

The fraction within the log is always greater than or equal to one, and only gets larger and larger with increasing $R$ and $\bar{r}$. But I don't know how I would go about expressing the rate of change with respect to the denominator of the ratio.
Hmm, I guess... what I'm looking for is more of a simplification, because that would be easier to look at. But I'm not sure what you meant by the fixed constant thing, can you elaborate?

 

[tommyxu3]

At first I'm not sure what you want also, and guess maybe you hope to get the ratio can be bounded with some cool number, maybe $Ce^R$... likewise haha.
For me the fraction cannot be simplified XD

 

[iScience]

For any set of values I've tested so far, the ratio has not exceeded the value 1. In fact i haven't found it to ever reach 1. I wanted to know what properties of a given set determined its closeness to the value 1. I hope this makes sense.

 

[tommyxu3]

What if I choose $r_i=100,r_R=1,R=2?$ Then the ratio is greater than 1 right?

 

[iScience]

$\{100\ ,\ 1\}$ yields...

$numerator: log_2{ \frac{ 101! }{ 100! \cdot 1! } } = log_2{ \frac{ 101! }{ 100! } } = log_2{101} =7$
$ratio: \frac{7}{102} = 0.0686$

(remember the numerator is logged by base 2)

 

[tommyxu3]

...My fault to miss the latter. I just saw the fraction. So the sum of $r_x$ is also concerned... Then it's harder to simplify...

 

[iScience]

I know I stated that simplification was my goal. But I suppose that's not the only way to achieve my "true goal", which I'm still getting closer to.. (sorry)

I postulate that there is a relationship between the numbers in the given set, that dictates its closeness to the value 1. I just want to figure that out..
even a pointer to how I might go about that would be of great help.

 

[Stephen Tashi]

given the set $\{r_i, r_{ii}, r_{iii}, ... , r_R\}$
where $r \ \epsilon \ \mathbb{Z}_+ \ , \ r_i \geq r_{i+1}$

As matter of vocabulary, if we were "given" that set of numbers, the ratio would have a single numerical value, so when you mention "bounds" aren't we treating the $R_i$ as variables and asking how to maximize or minimize the ratio? Can we start by solving the max-min problem for 2 variables ?