Criteria of considering relevance of the question to the domain of research topics

Question

Please look at
https://mathoverflow.net/questions/175757/conjecture-related-to-distinct-unique-permutations-of-digits. In my view, it is clearly a specific research question - it is not a general mathematics question ... What about the portion of the question where I am asking for references with regards to the topic of my question ? If though the question "denied" is indeed a general trivial mathematical question - would it be possible at least out of sheer politeness to give the references to the sources where the particular topic is covered ?

Answer 1

So following Eric's thoughts (which reflect my own), I would suggest (as a start) improving the notation and giving some examples of the basic concept. (It was only after looking at a relevant MSE post that I began to have a little idea what you were talking about.)

Listen: we mathematicians are human and sometimes need a little help. My impression is that you were trying to say too much in your post without giving a proper introduction. I see that you've posted before here with something on convergents to $\pi$ that attracted people's interest, and so it's possible that you're onto something interesting here as well, only people are having a hard time seeing what it is.

Please don't accept this answer as yet; I want to have a crack at rewriting some of it a little later. Or, perhaps better yet, please open up another answer box and try this yourself. If people can understand it, then maybe we can consider reopening or asking anew.

Edit:

Ah: the question was confusing to me because there was this mention of some curious coincidences involving ratios of certain "permutation pairs", and counting the number of such and relating this to the Euler $\phi$ function; this seems to me to be the real meat of the OP's concerns. But if the OP just wants confirmation of his empirical formula for one such special ratio [and that is indeed the concrete question I see], then if I understand the question correctly, the answer is very easy and elementary (and thus would not be appropriate for MO).

So: my current understanding is that the question is about a formula which computes the analogue in any base $r$ (the "radix") of the base 10 expression

$$\frac{9876543210}{1234567890} = 8.0000000729.$$

In other words, OP wants a closed form that calculates

$$\frac{\sum_{k=1}^{r-1} k r^k}{\sum_{k=1}^{r-1} k r^{r-k}}$$

As for references which deal with this, there must be many, but the techniques required can be found in any decent text which deals in generating functions as used in combinatorics and probability, such as Concrete Mathematics by Graham, Knuth, and Patashnik. The numerator can be evaluated as the value $f(r)$ of the function

$$f(x) = x(\sum_{k=1}^{r-1} x^k)' = x \cdot \left(\frac{x^r - 1}{x-1}\right)'.$$

The denominator can similarly be evaluated as the value $g(r)$ of the function

$$g(x) = x^{r-1} (\sum_{k=1}^{r-1} x^{-k})' = x^{r-1} \cdot \left(\frac{x^{-r} - 1}{x^{-1} - 1}\right)'.$$

So it just becomes a matter of a little tedious calculus at this point. I'll note that the author's $n$ here is $r-1$. His formula does check out for $n=9$, and I'm sure it's correct even without doing the tedious calculus.

Edit 2:

Probably the author does want more than the elementary derivation above. My guess is that he is interested in counting how many ratios of "pairs of permutations", where here "permutation" means an expression of the form

$$\sum_{k=0}^{r-1} \sigma(k) \cdot r^k$$

where $\sigma: \{0, 1, \ldots, r-1\} \to \{0, 1, \ldots, r-1\}$ is a bijection, are exactly equal to the special ratio whose closed form was computed above. He notes in the mathematics.se post (that he links to) that there are $7$ (yes, $7$) such distinct pairs in the base $r = 10$ case. This type of question really does seem curious and of potential number-theoretic interest, although I haven't thought very hard about it yet.

Answer 2

Some numerical experimentation (which could be considered recreational mathematics) has led to some curious observations which I don't properly understand but which might have number-theoretic import. I am hoping experts here can shed some light. If this is not considered appropriate for MathOverflow, then I apologize, and I ask for advice on a more suitable place to ask.

Among recreational enthusiasts, it has often been noted that the rational number

$$\frac{9876543210}{1234567890}$$

is very close to being an integer (the exact value is $8.0000000729$). What seems to be less well-known is that there are a number of such fractions, in which each digit appears exactly once in the numerator and denominator, with the same value as above, provided that we allow $0$ to be a leading digit:

$$\frac{7901234568}{0987654312}\;\; \frac{6913580247}{0864197523}\;\; \frac{4938271605}{0617283945}\;\; \frac{3950617284}{0493827156}\;\; \frac{1975308642}{0246913578}\;\; \frac{0987654321}{0123456789}$$

Each of these has the same value $8.0000000729$, or $\frac{109739369}{13717421}$ as a rational number in reduced form.

In more formal terms, there are thus $7$ pairs of permutations $(\phi, \psi)$ on the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that the base 10 expansions

$$\frac{\sum_{i=0}^9 \phi(i) \cdot 10^i}{\sum_{i=0}^9 \psi(i) \cdot 10^i}$$

all yield the same value $\frac{a}{b}$. Some extensive computer experimentation, courtesy of R. Cano and described in detail in a Mathematics Stackexchange thread here, suggests that if we define two permutation pairs $(\phi, \psi)$ (with $\phi \neq \psi$) to be equivalent if they yield the same rational value as displayed above, then $7$ is the size of the largest equivalence class, and this size is attained just for the value $\frac{a}{b} = \frac{109739369}{13717421}$ or its reciprocal.

My questions have to do with what happens in other bases besides base 10.

Question 1: For other bases $r$, consider permutations $\phi$ on the set $\{0, 1, \ldots, r-1\}$, and again define an equivalence relation on pairs of permutations $(\phi, \psi)$ (again with $\phi \neq \psi$) where two such are equivalent if they yield the same rational value $$\frac{\sum_{i=0}^{r-1} \phi(i)\cdot r^i}{\sum_{i=0}^{r-1} \psi(i)\cdot r^i}.$$ What is the size of the largest equivalence class? Calculations for the cases $r = 2, 3, \ldots, 10$ seem to yield, respectively, the sizes $2,2,3,3,5,3,7,5,7$. This looks suspiciously like one of the OEIS sequences here, all having to do with Euler's totient function.

Question 2: Which rational value $\frac{a}{b}$ represents this largest equivalence class? Putting $n = r-1$, my conjecture is that
$$\frac{(n^2 - 1) (n+1)^n + 1}{n(n+1)^n - n^2}$$ yields the maximum number, although there could be more than one such fraction. (Note that this is just a closed form expression for the base-$r$ expression $\frac{n\; n-1\; \ldots\; 1}{1\; 2\; \ldots\; n}$, analogous to $\frac{987654321}{123456789}$.)

There are other questions that could be asked; more can be found at the Mathematics StackExchange link given above. But this would be a good start for me.

(End of suggested edit. What appears below is an earlier version of the question.)

As an introduction, consider, for example, the set S10, containing all possible distinct permutations of all digits in the base 10: 1,2,...,8,9,0

One could see that the one pair (P) in that set {9876543210,1234567890} yields the ratio 9876543210/1234567890=8.0000000729...

Then one could find another pair in this set (S10), which gives the same ratio

7901234568/(0)987654312=8.0000000729...

Then the question arises whether there are other pairs (P) in this set (S10) ,which have the same ratio ...

Another question which arises - is there only one ratio in the set (S10) for which multiple pairs exist ?

And, finally, same questions could be extended to distinct permutations in other (than base 10) sets (Sn).

Also (as shown later) it appears empirically (based on computer program results for bases from 2 to 10) that the value of the ratio could be expressed as

(1)$$(n^2 (n+1)^n-(n+1)^n+1) / (-n^2+n (n+1)^n+(n+1)^n-n-1)$$ for n=1 ... infinity, where n = r - 1 and where r is the radix of the of the base.

For example, if one will take the base 10 (r = 10) then n = r - 1 = 9 If then one puts n = 9 into above formula - he will get 8.0000000729

So now I will try to generalize my questions as following:

According to an exhaustive computer program written for numerical bases 2-10 (which searched for the maximum number of pairs with the smallest, less than "n", but wherever is possible greater than 1 ratio(R) ), it was discovered that within an entire set of distinct permutations in the covered range, pairs (P) of permutations, which satisfy above stated conditions can be found, and that the number of such pairs is equal to: {2,2,3,3,5,3,7,5,7,...}. If one would consider the latter as an integer sequence - it might be (according to OEIS A039649, A039650, A214288) related to phi, which is the Euler totient function .

I have derived the above empirical formula (1) for the values of the ratio (R) for pairs in the given base (based on conditions as defined above): This formula (1) could be also expressed as A221740(n)/A221741(n),
where A221740 and A221741 are OEIS's integer sequences (submitted by me) to cover the values, generated by the numerator and denominator (correspondingly), which appear in the formula (1).

Gerry Myerson rephrased my question as:
given "n", find integers "a", "b" such that there exists "k" (greater than one, if possible, but less than "n"), so "ka" and "kb" use all n "digits" exactly once when written to base "n (with a leading zero permitted).

In Gerry's terms "n" is what I call radix "r", {ka, kb} is what I call to be a pair (P). What he defines as "k", I call ratio (R) - it could be expressed as k=i/l, where both "i" and "l" are integers. For each "n" the value of "k" is different. As empirical formula shows, both "i" and "l" (and therefore "k") are functions of "n". In each covered base "n" (from 2 to 10) several pairs, satisfying that "n" specific ratio ("k" or "R") were found - the number of pairs for given "n" is also a function of "n".

Based on obtained results two following conjectures are made:
1) Every complete set of all distinct permutations in any numerical base (radix) r contains some prime number of pairs (P) with the ratio (R) as defined above.

2)The ratio (R) could be calculated by the (empirical) formula (1).

If anyone has references already covering this particular topic - please provide them - I will appreciate such reference.

PS My analysis of results on this issue are described in
oeis.org/A212958
and in
https://math.stackexchange.com/questions/210578/permutation-identities-similar-to-7901234568-9876543210-cdot-1234567890/283117#283117

Could someone provide analytical proof or disproof of my conjectures stated above?