A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture).

(The question was first asked at math.SE, where (even after 2.5months and 2 rounds of bounty) there is only one special-case-solution and one additional insight on concavity. So I'm hoping to get some more insights here).

## The problem

There are $n$ different objects $A_1,...,A_n$, and there are sets containing $m$ different $A_i$s: $C_i=(A_{i_1}, A_{i_2}, ..., A_{i_m})$. There are $i_{max}=\binom{n}{m}$ different combinations $C_i$. Each combination $C_i$ has a probability $p_i$ (with $\sum_{i=1}^{i_{max}} p_i=1$).

**Defining the function**

For a given pair of objects $A_k$ and $A_l$:

- $f_1(k,l)$ contains all probabilities $p_i$ of the sets $C_i$, which contains both objects $A_k$ and $A_l$.
- $f_2(k,l)$ contains all probabilities $p_i$ of the sets $C_i$, which contains either object $A_k$ or $A_l$ (if it contains both elements, we add $p_i$ twice).
- $F(k,l)=\frac{f_1(k,l)}{f_2(k,l)}$

With that, we get the main-function $$D^{(n,m)}=\sum_{k=1}^{n-1} \sum_{l=k+1}^{n} F(k,l)$$

What is the maximum of $D^{(n,m)}$, given that the sum of all probabilities $p_i$ is 1?

**Alternative notation**

As pointed out by Wolfgang in a comment, the function $D^{(n,m)}$ can be written in a more intuitive way by the form: $$D^{(n,m)}=\sum_{i<j}\frac{\sum \{p_I|I \ni x,y \}}{\sum \{p_I|I \ni x\} + \sum \{p_I|I \ni y \}}$$ where $I$ denotes any m-subset of [n].

## Special cases

**n=2, m=2**

This is a trivial case. We have two objects $A_1$ and $A_2$, and only one set of combinations $C_1=(A_1,A_2)$ with $p_1$.

Thus $f_1(1,2)=p_1$, $f_2(1,2)=p_1+p_1$. This leads to $D^{(2,2)}=F(1,2)=\frac{1}{2}$.

Every other case with $n=m$ can be solved easily by $D^{(n,m)}=\frac{1}{n}$

**n=3, m=2**

This case is simple (but not trivial) and I found a solution:

We have n=3 objects $A_1$, $A_2$ and $A_3$, and combinations $C_i$ of m=2 objects $C_1$=($A_1$, $A_2$), $C_2$=($A_1$, $A_3$), $C_3$=($A_2$, $A_3$), with $p_1$, $p_2$, $p_3$ respectivly.

For k=1, l=2 we have $f_1(1,2)=p_1$ (because only $C_1$ contains both $A_1$ and $A_2$), and $f_2(1,2)=2p_1+p_2+p_3$ (because $A_1$ is contained in $C_1$ and $C_2$ and $A_2$ is in $C_1$ and $C_3$).

So we get $$D^{(3,2)}=F(1,2) + F(1,3) + F(2,3) = \frac{p_1}{2p_1+p_2+p_3} + \frac{p_2}{p_1+2p_2+p_3} + \frac{p_3}{p_1+p_2+2p_3}$$ A maximum can be found easily (due to normalisation of $p_1+p_2+p_3=1$): $$D^{(3,2)} = \frac{p_1}{1+p_1} + \frac{p_2}{1+p_2} + \frac{p_3}{1+p_3}$$ so each term can be maximized individually, which gives $D^{(3,2)}=\frac{3}{4}$ for $p_1=p_2=p_3$.

**n=4, m=2**

We have four objects $A_1$, $A_2$, $A_3$, $A_4$, and six combinations $C_1=(A_1,A_2)$, $C_2=(A_1,A_3)$, ..., $C_6=(A_3, A_4)$.

Therefore we get: $$D^{(4,2)} = \frac{p_1}{2p_1+p_2+p_3+p_4+p_5} + \frac{p_2}{p_1+2p_2+p_3+p_4+p_6} + \frac{p_3}{p_1+p_2+3p_3+p_5+p_6} + \frac{p_4}{p_1+p_2+2p_4+p_5+p_6} + \frac{p_5}{p_1+p_3+p_4+2p_5+p_6} + \frac{p_6}{p_2+p_3+p_4+p_5+2p_6}$$

I was not able to find a method for proving a global maximum.

**n=4, m=3**

We have $C_1=(A_1,A_2,A_3)$, $C_2=(A_1,A_2,A_4)$, $C_3=(A_1,A_3,A_4)$, $C_4=(A_2,A_3,A_4)$, which gives

$$D^{(4,3)}=\frac{p_1+p_2}{2p_1+2p_2+p_3+p_4}+\frac{p_1+p_3}{2p_1+p_2+2p_3+p_4}+\frac{p_1+p_4}{2p_1+p_2+p_3+2p_4}+\frac{p_2+p_3}{p_1+2p_2+2p_3+p_4}+\frac{p_2+p_4}{p_1+2p_2+p_3+2p_4}+\frac{p_3+p_4}{p_1+p_2+2p_3+2p_4}$$

This case can be simplified aswell, similar to $n=3,m=2$ case to $$D^{(4,3)}=\frac{p_1+p_2}{1+p_1+p_2}+\frac{p_1+p_3}{1+p_1+p_3}+\frac{p_1+p_4}{1+p_1+p_4}+\frac{p_2+p_3}{1+p_2+p_3}+\frac{p_2+p_4}{1+p_2+p_4}+\frac{p_3+p_4}{1+p_3+p_4}$$

but I'm not able to find any further method to calculate the maximum.

## Conjecture

The two cases I solved had a maximum at equal $p_i=\frac{1}{\binom{n}{m}}$. Furthermore, the function $D^{(n,m)}$ is very symmetric, so I expect that the maximum is always at $p_1=p_2=...=p_i$. Numerical search up to n=7 confirms my expectation (but I'm not 100% sure about my Mathematica-based numerical maximization).

## Questions

- How can you prove (or disprove) that the maximum for $D^{(n,m)}$ for arbitrary $n$ and $m$ is always at $p_1=p_2=...=p_i$?
- Is there literature on similar problems or is this function even known? Has the similarity to the Shapiro inequality some significance or is it just a coincidence?
- Is there a better (maybe geometrical) interpretation of this function?
- Can you find solutions for any other special case than $n=m$ (always trivial) and $n=3,m=2$? (See solution for $m=n-1$ and $m=2, n=4$ at math.SE)